2 Questions (spoiler alert!)

I just watched the film for the first time, and two things puzzled me. Maybe I wasn't paying close attention but...

When we find out that Paul is Rick's son, how come they have two different last names?

And:

When Rick's other son, the abductee Nick, says "Hey, didn't you used to come around & play with us when we were kids?" I think Rick says something like "We didn't want to impose". So I'm assuming we're talking about two different families, both headed by Rick? Who's Nick's mom (it can't be the same as Paul's sick mom, can it?)? And how did Nick not know he had a half brother?

Sorry, I guess that's more than two, but... I'd appreciate some answers! 