1. Gmail uses Labels not Folders.
2. Gmail will allow you to sync other accounts within Gmail. So
John.Doe@someotheremail.com can be retrieved into your Gmail Inbox. Worked for me.
3. Gmail allows you to "Label" the incoming mail from the other account specially so you can jump to that Label and read only email from that account. Just like Outlook filters allow.
4. Gmail will allow you to "Send as" so when replying people still will see your other email address in the "From" field. Gmail will even use the other accounts servers if you wish.
I just tried this and I love it. Now I can get my Gmail and other accounts in one place on my desktop and phone. Hope it works for you.