Depending on your phone / android version / launcher this cab mean different things I guess.

On my phone (MIUI) a dot like that means this is a new app, after you launch it a few times using that icon (using icon, not by other means like jumping into app from a notification or via opening a link) that dot disappears.

Also, I've noticed that when I type, not only do some things get underlined in red (I think it's for misspelling), but also in blue. Any idea what that means?

Red is a word that doesn't exist in the keyboard's dictionary (and likely a misspelling, but it sometimes doesn't have everything). Blue is a grammar check, and it's letting you know that your grammar might be incorrect (missing a word, verb doesn't agree with the subject, wrong word chosen, etc.). It's not perfect, but it may help you find mistakes you didn't intend.

I don't know what the green dot means. Sorry.

On Android, app just updated.

noticed the green dot as well and panicked - first few search results said that's an indicator someone's recording, either via mic or camera; checked the permissions log, no help there. haven't seen it after that.