Hi Dan, I have Android / Chrome here and it seems to work fine. Everything you say is by design, except it's supposed to shrink the chat bar when the keyboard comes out.
Do a full page refresh, it might clear up.
Seems obvious but I must ask... are you using the phone vertical or horizontal? Vertical is the only way you can see both keyboard and chat.
Try switching back to the stock Google keyboard if you're using something else.