I've noticed when you read the chat archives while people are actively posting, that the archive page contents are continuously changing.
For example if you are reading page 3 for a while, and in the meantime there have been 10 new posts to the chat, then if you click page 2 you will have a gap of 10 posts in the archive that you don't see.

The workaround is to simply refresh page 3 and check for new lines at the top before going to page 2 so you don't skip over any posts.

Do I automatically get signed up for all the forums I post in?

Yes, I think that's the forum default setting, you can change it here-

User CP > Edit Options > Default Thread Subscription Mode