USB port broken on a rooted phone

Depends on how they feel the day it arrives, but many manufacturers consider rooting to totally void all warranty. And, as assinine as that policy is (the only way rooting could have caused that problem is if you rammed the cable in when you were rooting it - which has nothing to do with rooting), the way the warranties are written, they're within their rights.

Unroot it if you can (search for methods). Then get authorization to send it in. DON'T keep pushing the connector to make it charge. Every time you do that, you make a little spark on the board, which chars a tiny spot on the board. Do it enough and you turn a "touch the contacts with a soldering iron tip" job into a "replace half the phone" job. And the more they have to do, the more they're going to look for a way out - like that tiny sign that the phone had been rooted. A loose connector is a goof on their part. They're likely to fix that and hope you still buy products from them.