Two bugs and a request
 

Bug 1: The TNT does not harm enemies when you are hit yourself and are invincible (flashing animation). It does explode, but the enemy keeps walking. It works fine when you are not invincible yourself. Looks like the invincibility is carrying over to the enemies?

Bug 2: When warping to a target in the air and the fall animation is showing, other events don't seem to cancel the fall animation. For instance if you buttstomp while falling, you buttstomp using the fall animation. If you fall onto a rope (battle1, at the bouncer upgrade), you will hang off the rope with the fall animation showing. Jumping from there will not cancel it either, only when standing on the ground, the animation is cancelled/

Request: Jazz 2 automatically pauses if focus is lost from the game window. Would it be possible to have an option to keep the game running in the background?

This is intentional behavior for TNT. I wouldn't mind changing it, it's not intuitive and TNT doesn't need to be worse than other weapons, but I'd want to see community consensus.
This has always been the case, probably not intentional but very cute. (It did get fixed for sucker tubes a while back, at least.) I could potentially support fixing the buttstomp interaction but I'd really rather not fix the vine interaction, it's like you're hanging from your feet.

TreyLina has previously suggested a possible compromise for the vines where you mostly get to keep the fall animation, but lose it if you shoot upwards from the vine.
Fully possible--this is how networked multiplayer works--but it doesn't seem like it'd obviously be a good idea? Generally if you don't have focus on the game, that's exactly when you do want it to be paused. Easy chance for enemies to attack you. Maybe it could be connected to the "Always on Top" window setting?