I, and many others, spend a lot of time answering questions in the chat.
This however doesn't get reflected on our reputation on the main site.
Can there be method for awarding reputation in the chat ?
Unfortunately that's oftentimes counterproductive.
If it is a question that's reasonably fitting for the main site, that's where it should be asked. That way, a) it also helps other people with the same problem, instead of being buried in a chat transcript, b) it discourages the laziness of those people who don't care to do the minimum amount of work to phrase a useful question, and c) guess what – you can get reputation for answering it!
If it's not a question that fits on the main site, then why should you gain reputation by answering it?
No. SE is Q&A site where users ask and answer the questions. Chat is a feature to discuss our issues in details. If user has any question which is fit to the FAQ of particular site, he can ask it directly on the site. Chat is localized because not all users use chat and not all users enter in the same chat room.