If not, at which point did the war first become impossible to win?
The defeat of the Third Reich was almost certian once Hitler and the Nazi party solidified their power over Germany.
It is important to understand that despite the atrocities they commited, the Nazi were not crazy. In fact if you come to terms with the premises of their ideology, they acted quite rationally on those beliefs. So some background information on where those beliefs from is in order.
Germany before the Nazi came to power, were in terrible condition economically. While it had a strong industrial base, Germany was over populated, and resource scarce. The powerful Germany industry was heavily dependant on imports of a wide range of resources from copper and textiles, to oil and rubber. But probably most importantly: food. Germany's population density was high with only a small amount ariable land per its population: (2.1 ha/farmer) compared to France (2.8ha/farmer), Britain(3.8ha/farmer), and the US(12.8ha/farmer) (Tooze). To pay for these imports Germany needed to export, largely finished manufactured products, chemicals, industrial machinery. Added to this were the war reparations Germany needed to pay; which again required foreign currency. Germany was a nation with a fragile economy, and massive debts.
When the great depression hit the global economy, the major trading nations started to pass protectionist trade laws to save themselves. That is to say they greatly reduced what they imported, while trying to maintain exports. However these beggar-thy-neighbour policies were implemented by all, so global trade sharply declined. This effect on Germany was pronouced. Without trade the economy nearly collapsed, and the repayment of war reparations had to be slowed.
It is from this desperate situation that the Nazi political party started to gather strength. The Nazi solution to the problems facing "the people without space", was to obtain colonies with the arable land and natural resouces the German people needed. Without these resouce rich colonies, Germany would forever be a second rate power. So where what this Lebensraum to come from? Eastern Europe is the obvious answer. Overseas colonies would be vulnerable to blockade by the British, French, or the USA. But the eastern nations, USSR, Poland, etc. were considered much weaker than the western ones. In the east Germany could have the space and resources it needed, while also being safe from any blockade by its traditional rivals.
It is for these reasons that war with the USSR had to happen. Invading the USSR was not a mistake, a rash decision, or an affliction caused by victory desease. It was a well thought out deliberate plan of action. Establishing lebensraum for "the people without space" was the only way Germany could retain its standing as a first rate power. The French Empire, British Empire, the USSR, and America, who did have these resouces would (Hitler and the Nazi believed) eventually leave Germany behind economically. Without the economic independance afforded by the lebensraum resouces and farmland, Germany would forever be at the mercy of its rivals.
To do this Germany would embark on a massive armament program. This program, rather than the "battle for work" social programs, was the spark that reignited the German economy (Tooze). From 1936-1939 Germany gradually re-engergized it's economy largely on a war footing (Tooze). While not yet having completed is very ambitious re-armament program, by 1939 Hitler had at his disposal a major fighting force. Germany was now ready to aquire its living space.
The problem now is two fold. First, to get to the USSR Germany would need to invade or otherwise annex the states between, mainly Poland and Czechoslovakia. However these countires had thier indpendance guaranteed by the French and the British. Second, any expansion of German power would be a threat to France and Britain who were likely to arrest Germany's expansion for this reason alone. Hitler and his high command were keen to avoid a two front war, the perennial german strategic problem. The solution came in the form of the Molotov–Ribbentrop Pact of 1939. With a non-aggression pact from the Soviet Union, Germany was free to fight a one front war.
After the stunning defeat of France, Britain refused to end the war. After the Battle of Britian, the British and the Germans came to a stalemate. Neither side had the air power required to force the issue. There is a line of thinking that Germany should have simply consolidated it's holdings at this point. Austria, the Czechs, Poland, France were all stunning victories. However, this didn't solve the living space or resource problems. Germany now under blockade by Britain was facing even more dire resource constraints. Worse support for Britian was growing in the US. The huge American industrial capacity was expected to soon bear down on the Reich. Germany didn't have the resources to face down both America and the British Empire. Further the USSR was continuing it's massive heavy industry expasion. Every year the Soviets were becoming more formidable, while Germany was slowly starving due to a lack of resources, especially grain and oil.
Time was clearly not on Germany's side. The sooner Germany invaded the USSR, the better the chances of victory. Hence the decision to launch Operation Barbarossa in 1941. The USSR while large in land and population, was considered backward and underdeveloped. Further there was an expectation that if a swift defeat of the Red Army could be delivered in the opening month of the war, that the Red Army would fall apart like the Tsar's army in WWI. At the start of war on the eastern front, this certianly seemed to be the case. A series of humiliating defeats showed that the Red Army was no match for the Wehrmacht.
Despite this, the USSR was not so weak as most people had assumed. The Battles of Khalkhin Gol against Imperial Japan provided the Red Army with veterain troops and commanders (most famously Zhukov). The Red Army equipment (like the T-34) was also suprising good. And despite the terrible leadership at the start of the war, the Red Army would become masters of operational warfare (Glantz). The Defence of Moscow, Operations Mars, Uranus, Bagration, and the defence at Kursk/Prokhorovka were all handled with considerable skill by Soviet generals (all Glantz). Additionally the USSR had already been expanding it's industrial capacity on the far side of the Ural mountains before the war (Montefiore). The massive relocation of Soviet industry from the Donetz region to Chelyabinsk, the evacuation of industry and critical workers from Moscow, and Stalins own papers, show that the Soviet Union was prepared to fight on even if its major cities in Ukraine and western Russia were lost (Montefiore).
This huge underestimation of fighting ability of the USSR, the opponent Germany needed to defeat more than any other, would be a fatal mistake. The failure to knock the USSR out of the war would mean most of Germany's workers (up to 85% of fit young men)(Tooze) would be fighting in Russia rather than building weapons to fight the still looming Anglo-American invasion. Germany would resort to pressed labour of much lower productivity, and never aquire the resouces or industrial strength to make victory possible (Tooze).
The Nazi plan to create Lebensraum in the east, was never going to work. British refusal to come to terms, the suprising resilience of the USSR, and the widely predicted entry of the USA, meant that Germany simply did not have the manpower, industrial capacity, or resouces to attain the living space called for in Nazi ideology.
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I'm almost remiss to answer this, because this is borderline /r/historicalwhatif in my opinion, but it is what I'd consider my specialty, so here goes:
Again, in my opinion, no, the defeat of the Third Reich was not inevitable from the beginning, so I'll explain why that is and then answer your second question.
In Hitler's Empire: How the Nazis Ruled Europe, Mazower details that a main goal of the Nazi regime was to consolidate supplies and soldiers from main allies (such as Italy and Hungary) and controlled satellite states (e.g. Vichy France). In itself, this was eventually to be unsustainable, but the turning point for Germany wasn't until its failed invasion of the Soviet Union (Operation Barbarossa).
Allow me to explain:
A commonly-held view, at least enough that I was taught it in school, is that Germany's major fault was opening a two-front war; until that point, they were performing rather admirably.
The issue wasn't nearly this simple.
/u/vonadler provides a miraculously well-written comment which explains somewhat how and why Rommel ended up invading Africa, which is an oft-ignored third front to Germany's war.
Germany is a country that is somewhat tactically disadvantaged, due to being smack-dab in the middle of Europe. At one point, it was being attacked by primarily England and the US from the West, England, the US, and France from the South, and the Soviet Union from the East, and at that point, the war was truly bleak, although unwinnable is probably inaccurate.
I'll focus primarily on the German invasion of Russia.
Operation Barbarossa opened up the largest front in military history, and was the source of 95% of all Wehrmacht casualties from 1941-1944, as well as 65% of all Allied casualties for the entire war. Germany's capturing of over three million Soviet POWs, and their treatment of them was probably great fuel for the Soviet atrocities that occurred in the back half of their counter-offensive.
Germany got all the way to Moscow, and could see the spires of the Kremlin, but they took too long. Even though the operation started on 22 June, but the Germans were only within fifteen miles of Moscow by 2 December, wherein, after already having incurred 620,000 wounded and 210,000 dead, Germany was caught by the infamous Russian winter.
Hitler himself would later blame Mussolini's bungling of Greece (and subsequent opening of a third front for Germany) for the Reich's defeat, but in all honesty, it was the losing several million men (4.3 million, by the most recent estimate, though I failed to find a citation for it) that really didn't help him.
EDIT: I just re-read how disjointed this comment is, and in part, that's due to how many moving pieces are in this question. You have to consider a list including but not limited to:
The list only continues. This is, as I said, not a very easy question to answer.
No, it was not inevitable from the beginning. Certainly, following the First World War, Germany was at a disadvantage given the restrictions put on it by the Treaty Of Versailles. However, Germany closed much of the "armament gap" in the 1930s (under the Nazis, but this process had its beginnings before their rise to power). However, the French in particular still, to most outside observers, possessed an army capable of holding their own against Germany, and the UK maintained their powerful navy's advantage.
A key event changed the situation though, in August 1939. After negotiations for an alliance broke down with The Western Powers, Stalin signed the Molotov-Ribbentropp Non-aggression Pact. This of course meant that Germany would not face a two front war (at least not for a while) as it had immediately in 1914. Still, things were not inevitable. While it would be difficult, the war still seemed winnable for the British and French. Their defenses on The Maginot Line were still formidable. Of course, as it turns out, the Germans ended up largely bypassing the line through the Ardennes defeated the French, and drove the British off the continent. There was nothing inevitable about this victory though--it turned out the German doctrine was superior (massing their tanks in formation, greater radio coordination between air and ground forces) but it wasn't as if the Germans knew that this attack through the Ardennes would be a success, it just seemed like a better alternative to repeating Schlieffen (attacking through Belgium). The speed and success of the campaign in the West astonished the Germans almost as much as it did the British and French.
It's hard to argue that by the end of the year 1940, and the beginning of 1941 the Germans weren't sitting in a good position. True, they did fail to win air superiority over Britain and drive them out of the war, but on the other hand, the British could not inflict catastrophic damage on Germany either to drive it out of the war. At sea, the British were losing more in the tonnage war than they could build. On land, when German forces engaged Allied the Gemrans tended to be successful.
Keeping these in mind there are several things that make 1941 the key year in which the war became, in hindsight, unwinnable for Nazi Germany.
The Lend-Lease Act, March 11, 1941--The US now gives billions of dollars free, or nearly for free, in weapons and supplies to countries fighting the Axis. Most importantly the USSR and British Empire. This ensures that the gigantic industrial might of the United States would now be against Germany.
Invasion of the Soviet Union, June 22, 1941--The the largest invasion of world history. This front will tie down more than approximately 2/3 of German troops for the majority of the rest of the war.
German advance on Moscow stalls on December 5, 1941--BARBAROSSA ends, German troops settle in for a protracted war deep in Russia and their strained supply lines are inadequate to the task.
The United States declares war on Germany, December 11, 1941--this makes official what has already been an undeclared war for months. FDR agrees with Winston Churchill on a "Europe First" strategy--Germany, the most important opponent, must be eliminated before Japan. The majority of US production goes toward Europe.
By the end of 1941 Germany finds itself outnumbered and outgunned, on multiple fronts. In 1941 The Big Three (US, UK, USSR) have a combined 4:1 advantage over Germany in GDP; in 1942 it is 4.2:1; in 1943 it's 4.5:1; in 1944 it is 4.7:1. It became a war of attrition the Germans simply could never hope to win.