At the dawn of World War I, the Great Powers were willing to go to war but also hoped that their would-be opponents to back down. Is it possible to rank which countries wanted war the most versus which ones wanted peace?

I'm not really going to rank them, I'll just go down the list.

Russia--Russia was in a complicated position prior to the war. On the one hand, they sought to protect their Slavic ally, Serbia, from Austro-Hungarian domination (and therefore preserve Russian prestige and a foothold of influence in the Balkans.) On the other hand they were not in any position to go to war in 1914. They were still somewhat reeling over their loss to Japan in the Russo-Japanese War years before, and the 'Great Military Program' (for modernizing and reforming the Russian Army) would not be completed at the earliest until 1917. Russia knew their size could be used to intimidate, but they also greatly feared the prospect of going to war (especially a war with powerful, industrialized Germany). On the whole though, it is clear the Russians for the most part did not want a war in 1914. During the July Crisis, they advised Serbia to "accept Austria-Hungary's demands without being humiliated too much" (Serbia agreed to 9/10 of them). Also, in the famous 'Willy-Nicky Telegrams' between the Tsar and the Kaiser, Nicholas II offers to take the dispute between Austria-Hungary and Serbia to The Hague, and Wilhelm II doesn't respond to this proposal.

France--While it is true that France sorely wanted Alsace-Lorraine back, I've never been convinced of the argument that they were itching for a war to do so. France, curiously if they indeed wanted war, advised the Russians not to provoke Germany continuously throughout the July Crisis. Another thing was that the French couldn't be absolutely sure of was that the British would enter the war at all, or at least that they would do so in time to help France. And with Russia possibly preoccupied in the east with Austria-Hungary, France standing alone in the west against Germany was a grim prospect indeed. And it was fairly well known among the governments that in an event of war, Germany planned to strike its hardest hammer blow first against France.

Britain--The UK had no intention at all of getting involved in the continental dispute at first. Indeed, they were more worried about the state of affairs in Ireland. The British urged that Germany should put pressure on Austria-Hungary, and this would be the easiest route to peace (obviously this fell on deaf ears). It was only when it became clear France would be drawn in, that the UK seriously took the prospect of war. The illegal invasion of Belgium by Germany gave Britain the public support to enter the war.

Austria-Hungary--They both hated and feared Serbia at the same time, and for some time had sought to destroy Serbia once and for all. In fact, they had planned to do it in 1912 and again in 1913--but Germany had told them the political conditions were not right. The reason was that the Austro-Hungarians tended to view Serbia as a destructive force of nationalism on their multi-ethnic empire. Actually, when the assassination happened, the document which would become the ultimatum they would issue had already began to be drafted. The assassination (of the archduke, who wasn't particularly liked) gave them a pretext for something they'd wanted for sometime, the destruction and annexation of Serbia.

Germany--Germany was the country that wanted a "general war" more than anyone else did. They were aware of the Russian military program, and they looked on with great suspicion. They were aware of Russia's huge population, and if the country were to modernize they knew they stood little chance, especially with their old enemy France ready to strike them in the back. Hence their leadership, Moltke the Younger, Falkenhayn, and Bethmann-Holwegg all agreed that the time to strike was now. The problem was that in order to get the various factions within their own population behind the war they needed Russia to look as an aggressor. When the assassination happened, and Austria-Hungary came to them for support, they saw an opportunity. They gave Austria-Hungary unconditional support for any move they made. Essentially it was saying that Austria-Hungary had a free hand to attack Serbia, and Germany would handle Russia if Russia tried to intervene to stop Austria-Hungary. But Germany knew that, in order to have their desired war with Russia, first they needed to take out France quickly--and then they could turn all their attention east (while Russia slowly mobilized their huge army). So essentially, Germany had no plan to go to war with Russia without also attacking France. Germany did hope to keep the British out of the war. The problem was that they had built a huge navy (ironically in an attempt to intimidate Britain) which had severely damaged the two powers relations, and had actually driven Britain from isolation into the Entente. Still, the British were not committed to going to war on the continent until (in an attempt to defeat France ASAP) the Germans invaded neutral Belgium. Belgium happened to be home to a key port, Antwerp, which the British thought of as quite threatening to their island's shores were the Germans to be in control of it. Germany underestimated Britain's commitment to Belgium's neutrality and their desire to have France remain a great power, which they wouldn't be anymore had Germany be left to defeat France as they had in 1871.

So in conclusion, I would say that the countries which wanted war more than any others would be Germany and Austria-Hungary.

Source: Europe's Last Summer: Who Started The Great War In 1914?, David Fromkin

Overall, I would have to simply say, no, it really isn't possible to say which countries were more eager or willing to go to war. But thats not why you came here! What we can do is try to narrow down the question to get at something productive and informative. Instead of saying 'which countries' we might ask instead how individual political or military leaders of the day viewed the positions of their countries in the coming war- what advantages or disadvantages they had going in. Or, we might ask how different governments envisioned a post Great War world, what they expected to gain for their countries or what more pessimistic leaders thought the costs might be.

To further complicate things, not everyone living in these countries either wanted to go to war, or even felt as though they really belonged to those countries. Many Czech inhabitants of the Austro-Hungarian Empire maybe wanted to go to war, but as a way to gain political independence for their people. On this basis, you could say that Austro-Hungary was very interested in going to war, but it wouldn't really be getting to the initial intent of the question.

The question of who really wanted the First World War, or who wanted it the most has occupied historians for almost 100 years now and to be honest, the closest thing that I have ever seen to a satisfying answer is "Everyone wanted it really badly," and that still doesn't get to the truth of the matter. Its just more complicated than that.