I'm not sure why but some users are having horrible problems with Gmail. Usually doing the nuke options will get it to work.
We can look at workarounds or look at the possibility of using Gmail as POP in Mail. You can archive most of your Inbox messages to Gmail's All Mail folder.
Another option is to look at third party email clients.
Postbox
http://www.postbox-inc.com/
MailMate http://freron.com
MailMate has easy setup and will import your accounts from Mail. (you would revert back to original Mail folder) Its IMAP only so if you have POP accounts this won't work for you.
You might like MailTabPro for Gmail $1.99. In Mac App Store (MAS). Nothing to download and easy access to your messages.
https://itunes.apple.com/us/app/mailtab-pro-for-gmail/id430252530?mt=12&ign-mpt= uo%3D4
There are a couple of others but right now, I can't remember names.
We can continue to look at your Gmail accounts to see what might be the issue. I would start with checking settings on Gmail and clean up labels.