I need a gmail expert

I need someone to help me sort out our gmail accounts. We have recently switched from pop to imap and have gone from one free gmail account to three and we are having all sorts of problems with emails appearing then disappearing and constantly being asked for our login password. I need someone who knows gmail in and out to help sort this out.

Problems we currently have:
Although we have an integrated inbox, sent and draft box, I hate the way the 3 accounts are separated in the folder pane. I hate the way the gmail folder system works which seems to give me folders I don't want but don't know if I can delete (starred???). Also what's with the archive folder that seems to have gazillion emails including 10 different version of one email as it was in the process of being written, I really don't need that once it has been written and sent it should go away.

What I ideally want:
One email account which should contain inbox, sent mail, drafts, junk and deleted. I also want to set up folders for client emails but they should all be in one place, say to find and viewable on all devices.

A possible solution could be to create an archive of all the old emails that we could download to our local computer for back reference and then forward all 3 free gmail accounts to one pro account? Open to suggestion on the best way to get this to work