Hi Anne!
Did you solve your issue?
About remark 1 I believe that there is no way to SnapEngage trigger the C4C when the customer trigger the chat link.
About remark 2 take a look at Note 1935691:Error: Unsupported Content Type is received
And about remark 3, CTI Integration is supposed to have a chat integration as well, as you can see in the documentation of Administrator Guide of 29/04/2016 on page 196.
Best regards