itch.io

While streaming you can minimise T.I.T.S and your broadcasting software (OBS/streamlabs/...) will continue to capture it regardless - that way you have access to your regular desktop. Hope that helps.

T.I.T.S isn't related to any capture software at all so I'd say no, why should it? If you want to use T.I.T.S with streamlabs simply setup a game capture source and let it capture the T.I.T.S application just like you would do with OBS or Twitch Studio or any other capture software.

HONK sometimes lists an audio input twice so it's possible you have the wrong of the two.

Also: no lipsync or other movement as long as the HONK window is active (learned that the hard way, selecting a different app or clicking on the desktop and it syncs, click on HONK and it stops so you can edit).

Properly support all possible F-Keys please.

It's okay to have support for combinations as we already do but CTRL+ALT+SHIFT+F1-F12 is pretty much used by a lot of apps.
F-Keys go up to F24 though, even if they're not on most keyboards they can be mapped using any number of utilities as well as by a streamdeck. 

Veadotube and OBS support those keys and the clash rate with default system- and application shortcuts is close to zero.

A little observation on this issue (I do have it as well with 3.x) - it only happens if the HONK window is fully covered.
An easy test to reproduce it is to have HONK and OBS on the same screen - everything works, then maximize OBS so it covers HONK and a moment later my model starts chattering.