Short answer: The US Navy was indeed engaged in combat against German U-boats in fall of 1941, before the attack on Pearl Harbor and Germany's declaration of war on December 11. These actions were part of the American "neutrality patrols" which had been initiated following the outbreak of war in Europe, but the situation escalated from 1939 to 1941. A full accounting of the diplomatic efforts between the United States and Germany is probably worth its own question (and beyond my knowledge). I can provide some context on the support that the US offered the Allies (specifically Great Britain) before Pearl Harbor. Suffice to say that it was well known prior to Pearl Harbor that the United States was not really neutral. She had picked a side and took increasing actions to support the Allies. This culminated with three high-profile incidents in fall 1941 in which American warships and German U-boats were firing upon one another, which led to one US destroyer being sunk and another badly damaged, with significant loss of life in both instances.
The long answer:
Diplomatic relations between the United States and Germany had deteriorated throughout the 1930s. America was troubled by the rise of fascism and human rights violations throughout the decade, among other issues. Franklin Delano Roosevelt's State of the Union delivered in January 1939 (well before Poland was invaded) did not mention Germany or Japan by name but was focused almost wholly on the international situation^1:
An ordering of society which relegates religion, democracy and good faith among nations to the background can find no place within it for the ideals of the Prince of Peace. The United States rejects such an ordering, and retains its ancient faith.
There comes a time in the affairs of men when they must prepare to defend, not their homes alone, but the tenets of faith and humanity on which their churches, their governments and their very civilization are founded. The defense of religion, of democracy and of good faith among nations is all the same fight. To save one we must now make up our minds to save all.
He also discussed the idea of neutrality, stating that the action or inaction of the United States shouldn't inadvertently assist aggressors:
At the very least, we can and should avoid any action, or any lack of action, which will encourage, assist or build up an aggressor. We have learned that when we deliberately try to legislate neutrality, our neutrality laws may operate unevenly and unfairly—may actually give aid to an aggressor and deny it to the victim. The instinct of self-preservation should warn us that we ought not to let that happen any more.
Roosevelt's argument about "neutrality" here is important. The US quickly declared that it was not joining the European war after Germany's invasion of Poland in September 1939. However, the United States would put its thumb on the scale in favor of Britain and France. Roosevelt urged Congress to repeal legislation passed earlier in the 1930s that was designed to keep America neutral, resulting in a new "cash and carry" policy. This allowed Britain and France to purchase weapons and war material from the US, but only if they paid in cash and agreed to transport the purchases on their own vessels.^2
The United States also instituted "neutrality patrols," in which American planes and warships began escorting ships in "neutral areas." This was presented at the time as a way for America to protect its own interests and remain out of the war, by guarding areas close to American ports and shipping lanes all the way from South America and into parts of the Atlantic. Neutrality patrols would later expand and play an important role in 1941.
We can pause here and make two brief notes. First, vats of ink have been spilled by those who debated whether Roosevelt's policies were actually intended to keep the US neutral or if they were designed to draw the United States into the war. That's a debate for another day. What is not really up for debate is that while the United States was not a co-belligerent in the conflict in 1939 and 1940, it had certainly chosen a side (the correct side, of course). Second, the German U-boat fleet in 1939 and 1940 was a problem but not the danger that it became in 1942 and early 1943. The Kriegsmarine had a number of U-boats when war broke out in 1939, but was much smaller than the fleet of U-boats operating in later years. Many of the U-boats that in operation at the time also lacked the range to operate in the open ocean. I don't think it would be controversial to say that the Wehrmacht and Luftwaffe had received more resources and were better prepared for the outbreak of hostilities in 1939 than the Kriegsmarine.
Back to your answer.
American public opinion on the war shifted significantly in 1940, especially after the rapid fall of France and the low countries left Britain alone and isolated. The United States agreed to the "Destroyers for Bases" deal with the UK in September 1940, transferring 50 mothballed destroyers (a warship that was an important part of convoy escorts) to the British in exchange for the ability to construct bases on some British holdings in the Caribbean and Newfoundland. The destroyers were older designs from the late 1910s, and many were in poor shape. We can let Royal Navy Admiral John Tovey take it from here:
"I thought they were the worst destroyers I had ever seen, poor seaboats with appalling armament and accommodation. The price paid for them was scandalous."
Nevertheless, it was an important step up in American involvement. These older destroyers (of the Wickes, Caldwell, and Clemson class) will play a role later in this story.
The next important evolution was the "Lend-Lease" policy that Congress approved in early 1941. The British economy (and others who benefitted from Lend-Lease) was strained by the war, and no longer had to pay for its goods in cash. Resources and weapons could be sent to the British to be returned after the war or paid off at a later date; in practice, most of the surplus war equipment the US built and sent to Britain, then scrapped after the war. It wasn't as if the Americans needed lots of war material to be returned in 1946, they had plenty as is. The French government in exile was also included in Lend-Lease; the Chinese were added soon and the Soviet Union was added later in 1941 after Germany attacked.^3
If the United States was going to be building war materials and sending it to Britain, including on some American merchant ships, it was also time for American ships to defend those shipments. Neutrality patrols grew and by April 1941 included areas as distant as Greenland (a Danish territory the US sought to keep neutral after Denmark was captured by the Germans) and the Azores; by summer 1941 it included Iceland.^4 The scale and mission set of neutrality patrols both grew; American warships also began reporting the position of German submarine sightings or sonar contacts to the British.
This was an increasingly dangerous situation. American warships were now operating far into the Atlantic Ocean in waters where German submarines are, providing protection to merchant ships. The US Navy ships were not allowed to engage German forces they encountered, but had the right to defend themselves if attacked.
A German U-boat sank a plainly marked American merchant ship, the SS Robin Moor, off the coast of South America in May 1941.^5 The U-boat permitted the crew to evacuate to lifeboats before it was sunk, and there was no loss of life. Still, this couldn't have been a mistake: It was a daytime attack on a ship that was clearly marked. This was not typical, however. German orders were strict about not engaging American ships; despite heavy American involvement, it was still better for Germany to keep America out of the war. This was an unpopular order among U-boat commanders, who were taught to be aggressive. It was counter to their doctrine to exercise so much caution when acquiring a target.
This leads us to the three events that you're probably referring to in your original question.
1/2