Snails as an ammonia source would work for that purpose, but I've heard (no personal experience) that it can be difficult to find and remove snails (their eggs specifically) from things like plants and decorations. For that reason, I probably wouldn't rely on this source, because you should not have snails and axolotls together. When I cycled my first tank, I actually used 100% liquid ammonia that I think I bought off of Amazon. I just added a particular amount every 24 hours.
It's great that you have that filter media! It's much easier to maintain a cycle than to start one, but you will still need a source of ammonia in the new tank. Theoretically, I think you'd probably be fine just to put the filter media in the new tank at the same time as your axolotl as long as you keep an eye on the ammonia, nitrites, and nitrates. The bio-load will probably be different with the axolotl (they're little waste machines, haha), but with regular water changes and vigilance of the levels, you should be fine.
As for the bugs, what kind of bugs are you dealing with? What's the source? If the issue is caused by some kind of environmental problem in your home, I would work on resolving that immediately. Depending on what kinds of bugs there are, they may cause problems if your axolotl tries to eat them. That being said, I've gotten the occasional moth in my water. It happens. I just use a turkey baster to remove them as I see them and all is well. I do personally recommend some kind of lid, especially with a new axolotl and/or one in a new environment. They are known to jump out occasionally. As I said before, you don't necessarily NEED one, but I use screen lids for all of my axie babies, even though there's really only one I'm slightly concerned about, lol.
As for the vinegar issue, I would say that it's better to be safe than sorry. Ultimately, it's up to you, but I personally would empty and refill it. If you don't have a siphon or gravel vacuum and a 5-gallon bucket (or two), I would definitely recommend the investment! Makes it soooo much easier to do big water changes!