The title pretty much says it all.

Which weapons can you use while still maintaining invisibility?

The wiki only says that invisibility will disappear by using some specific weapons.
http://worms.wikia.com/wiki/Invisibility

And also, can you use all utilities while being invisible? I am mostly concerned by Teleport (can the blue flash be seen?) and rope (can the rope itself be seen by enemies?)

And thanks TEAM17 for an awesome game. So simple, yet so complicated and brilliant. Playable for all ages! :D

In general u can use all the tools and weapons that will not obviously reveal your position:

i.e.: if u use torch every body would know anyway were u are since u digging and you will be relealed. Different for chute and ropes and jetpacks. they cannot be seen neither teleport but remember 2 things:
1. camera will follow u during a teleport and center view on you. so if the map is big enough to scroll everybody will know u teleported at the center of the screen.
2. when u slide after a roping or slightly fall you will raise little dust clouds. those can be seen.

Of wepons u can only use indirect hit stuff like "double damage" (u know the cloud on your head) or steal etc.

http://worms2d.info/Invisibility

Same applies for Reloaded probably, excluding freeze and bungee and including that whole team teleport thing.

Edit: Reading again, I'm not sure about teleports now (there is no flash in the 2nd generation games). Best bet is to try it!