There's not really any way to explain why certain words are with chat in these situations, except that they've evolved through the internet age and are just how we say it, now. So there is no source or reference for me to give you, except that this is standard usage:
I am currently in a chatroom.
I am getting on Skype (or other chat provider).
I asked him a question on chat the other night.
I asked him a question on/over Skype (or other chat provider) the other night.
To speculate as to why these patterns came about:
It's likely that you are described as in a chatroom because it contains the word room, and if we take the abstract concept of a chatroom and consider it a physical object, then you would always describe yourself as in a physical room.
You get on/log on to a chat provider similarly to how you log on to any other website; it's just how it's phrased, I don't have any speculations as to the etymology.
Perhaps we talk on chat because while on chat we are online, or because the text we type appears on the chat window.
When referring to the chat provider by name instead of the generic chat in general, perhaps we can use over similarly to how we can say over the phone; we're imagining the chat provider as sending our messages across, or over, their network.
At any rate, the above is just speculation as to why these usages are prevalent. Just stick with the rules and you should be fine, I doubt there will be a pop quiz on the origins! :)