I asked this exact question almost two years ago here and received a concise response from /u/Erusian, albeit one that shoots down any interesting interactions one would hope for. Obviously there may be more to it, so I hope this does not stifle others to respond.

Since there's been so much interest in this post, I'm going to elaborate a little. First, I'd like to begin by saying I do have the distinct advantage of speaking English and German but that I do not consider myself a central European expert. So I will do my best but it's possible I will miss something a more seasoned central European scholar would not.

The United States first made contact with non-English Europeans through three commissioners in 1776. They did not visit Germany but did have contact with Germans, mainly North Germans through the French. There was some sympathy in the Empire to the American cause, mainly because it was seen as anti-British and most of the Empire (though not all of it) had been on the anti-British side of the Seven Years War. And Britain's biggest German ally during the war, Prussia, felt it had been snubbed and frustrated in its territorial ambitions by Great Britain. Thus the Empire was not generally pro-American exactly but they were anti-British. There were some exceptions: Hesse took British money, Hanover remained loyal. Some other minor principalities remained stalwart British allies. But for the most part the Empire was generally pro-American. Several principalities, for example, refused to let British troops pass through them or recruit there. However, this did not reach the level of a national relationship. The first treaties were only signed after the American Revolution.

Austria (and therefore the Empire as an organization) declared neutrality during the American Revolution but took some actions to inconvenience Great Britain. It was joined in this stance by Russia, who agreed to cooperate on their foreign policy in this matter. When John Adams came over to try and negotiate terms with the British, the Austrians and Russians supported a negotiated peace. But this amounted to little more than diplomatic pressure and neither John Adams nor the British considered the proposed terms acceptable.

Austria and several other German principalities signed treaties of amity and commerce with the United States in the 1780s after the war was over. But these were basically recognizing the United States' independence and setting terms for trade. The United States did not send a diplomat to the Emperor. The US did sometimes have a consul whose primary responsibilities were trade based. The consul did not reside at the Austrian court, he resided at Trieste, a trading port. He was not in an envoy, minister, or ambassador. There were multiple treaties but they were almost all commercial. Prussia had two before the fall of the Empire, for example.

Revolutionary France declared war on Austria in 1792 and a Polish revolt in 1793 led to the Third Partition of Poland and the War of the First Coalition. American thoughts on the First Coalition were split but dominated mainly by considerations of Britain and France. There's a fairly clear pattern: American writers were most concerned about European countries with territories close to the United States. There was relatively little mention of Austria. American attitude towards the partition of Poland was more united: America condemned the partition but did nothing else. To be fair, they really had no levers to pull.

Ultimately, the United States remained neutral in the war until 1812. But the Empire dissolved in 1806, six years prior. The Third Partition of Poland would later inform the first major diplomatic row between the two during the Revolutions of 1848 (America very strongly supported the Hungarians and other ethnic minorities). But by that point the Empire had long dissolved.

In general, the United States did not have a diplomatic presence in Germany until the 1830s, putting a diplomat in Prussia in 1835 and Austria in 1838. (There was also a brief period, from 1799-1804, when John Quincy Adams served as a Prussian diplomat. But the thirty year gap makes this rather temporary). Meanwhile, Prussia didn't have a diplomat in the United States until 1825. When the diplomat was reassigned in 1830 they didn't replace him for four years, finally establishing a permanent presence in 1834. Austria posted a diplomat in 1838, the same year the United States did. Other than that, the only Germans to post ambassadors to the United States were a couple of German trading cities. They mostly used them to angle for better trading rights. In fact, this was true of the United States, Austria, and Prussia too. They were all primarily concerned in securing trade.

To answer your questions directly:

Were the State's formally reconized by the HRE?

Yes. At the latest by 1797 when they allowed the US to establish a trade consulate, but more realistically after the peace in the 1780s. Recognition by the HRE meant recognition by the Emperor, ie, Austria. They did not accredit an ambassador or send one: no US diplomat was ever recognized by the Emperor or Imperial diplomat by the President. But this just means the two countries didn't have diplomatic relations. It was not a comment on the legitimacy of the state the way it is today.

If so, did the two consider each other rivals, allies, or was the Empire too busy collapsing/America too isolationist at the time?

They just didn't have a lot of areas of common interest. Austria and all the other HRE states had no American territory to speak of. Their relationship, such as it was, was almost entirely navigated by proxies. The Austrians didn't have an opinion of the Americans during the American Revolution but they had an opinion of Great Britain, which informed their opinion of Americans. The Americans didn't have an opinion of Austria during the War of the First Coalition but they had an opinion of France which informed their opinion of Austria. Mostly, the two saw each other as trading opportunities. This led to increasing contact and diplomatic exchange. The first real time American diplomacy was about Austria itself was during the diplomatic row of the 1840s. The United States was deeply upset at Austrian actions during the Revolutions of 1848 and some even wanted to try and intervene. But by that point the Empire was long gone.

From The Fall of Poland in contemporary American opinion, American Foreign Relations Since Independence, Recognizing States: International Society and the Establishment of New States since 1776, and Divided Union: The Politics of War in the Early American Republic.

There's a great answer here by /u/Erusian