I don't think the XYZ affair would have been handled too differently. We'd probably see different names involved, but ultimately, Adams himself didn't do much to involve himself in the situation, I think Washington would have done even less. The details of the affair would have been released to Congress, per OTL, and really Congress, the Federalists at least, were already pushing for war at this point. I think Washington would have been quite adamant about avoiding war, but I think we'd still see a quasi-war. Since there's no declaration of war, I think it's a little more agreeable to Washington.
When he dies, Adams does not become president in his own right. People like to think of this time as one of good feelings between all parties involved in the affairs of the nation, but the earliest years were some of our most volatile and had the most potential to see serious damage done to the nation for simple political reasons. The Republicans at this time would have flipped fecal matter at the prospect of Adams becoming President without an election. The matter is unprecedented, but the Constitution is a fresher institution than when this occurred in OTL, so our political culture isn't as developed and people have larger and less tested worries about what the President might be capable of. Also, Madison (the famed author of the Constitution) being a Republican would likely be against Adams becoming more than acting President and he'd probably garner a lot of support in that position.
As for how long future Presidents stay in office, I think it depends on what the next President does. If he defers to two terms and makes a deal about Washington staying for so long because, well, he's friggin Washington, I think we could still end up with an unofficial two-term limit. But the door is definitely open for Presidents to run as often as they like, two, three, however many terms they like, even if they're voted out of office at some point (didn't stop Cleveland, after all).