As for the mail. I too endorse Thunderbird. It handles multiple accounts for IMAP and/or POP and can do either separate inboxes or a combined inbox, but no Metro UI.
I realize Windows mail isn't the greatest app in the world. . .but it DOES natively support gmail accounts.
And for the O/P, the latest editions of chrome stay running even when the browser is closed [with permission] to transmit your hangouts/G+, etc.
I can't help with three separate hangout accounts tho. You log into gmail with that account and it's just there.
I'm not even sure how you set up three separate "hangouts accounts" in Android