Just loaded it up by itself, and saw this:
http://imgur.com/KDV4li5
ChatFrame2 layout seems to be quite broken. There's no visible tab or any other way to interact with the frame, and the button frame next to it doesn't match the frame height.
For the actual questions:
1) Could not reproduce, with or without other addons, and I don't see anything in your code that should be causing it. Try clearing your cache.
2) This is happening because the function you use to skin the tabs ("Tabify" in flash.lua) doesn't accept an argument telling it which frame to skin. It just loops over all the normal chat windows and skins their tabs. You need to either (a) call "Tabify" from your "skin" function instead of calling it at PLAYER_LOGIN or (b) set up another hook to call "Tabify" when new chat windows are created.
On a side note, while your "skin" function does correctly reference the frame passed into it, it's doing so 10 times, because it's looping over 1,NUM_CHAT_WINDOWS and running the code for each of them. Get rid of the loop wrapper.