If someone leaves a message in their turn, will I see it if I skip watching their move? Seems I never get any messages if I skip... If so this should be adressed! Shouldn't the skip "pause" at any messages?

When the message is send before any move, you will get the message before you can click skip.

I don't know what happens with message's during moving phase..

I wonder also, if I send a ingame message during my round, and I rewind all, does the message goes away too?

Yes it does.

I agree with Tryptamon. Pausing for messages when skipping the turn is a great idea.