What is a word for someone who has a hard time denying other people's requests and who speaks politically correctly because he will feel guilty making someone feel offended?
For the first (someone who has a hard time denying others' requests), I'd say pushover, milquetoast or doormat. A less critical word to describe such a person would be softy.
Softy could probably apply for the second definition as well. I think the best way to express the idea that someone is serious about being politically correct is by using those words (or by shortening them to the well-known acronym PC) - they have a pretty specific meaning. But you could also use a word like inoffensive or even spineless.
I think the words available to describe a pushover are far more satisfying, and you could generally get away with just leaving the PC part out and still reflect the person's general character. If you feel the need to emphasize both, you could combine the ideas with something like He's such a PC pushover.