Is there a way to skip the videos in beginning of co-op play and the videos between hub and chamber?
You are at a loading screen if you can't see each others messages and in-game when you can.
During the video in the beginning of co-op play you can talk, which means that you are in-game. You can use this time to chat with your friend in advance, but if you play regularly with someone you know this can indeed be annoying. While trying around yesterday I have noticed that you can influence the settings of the co-op game.
Thus the video in the beginning can be skipped by changing the host time scale:
host_timescale 16(if it works, it should up the video by about 16x, shouldn't take long)
host_timescale 1(Set it back, or you'll be unable to play the game properly)
These server-side settings are adopted to the other player, so you'll see them executed twice.
The videos between the hub/chambers can't be skipped, because they are loading screens.
Notice: You can not get achievements for that game anymore...