Hello,

Is it possible to have it insert user is typing somewhere in the module, so you know when the other person is going to respond?

This is possible but it will give high load on your sever resources .

where could i do this?

off topic question, more for the future, but how easy would this to be to run the server for the chat separately than the website server?

It would require a lot of tweaking, I would recommend you to wait rather than tweak, as it will be added in the upcoming major version anyway.

Running the chat from a different server(but using the same domain) shouldn't be a problem, it may require some tweaking though.
The next major version will include support for cross domain too.