(I hope everyone is a fan of thread revival)
Yes, the characters do hug you and interact with you. It's part of their training and their routine for mingling with the park crowd, especially children. Some of the characters will even pick up and hold your children if permitted, although they're usually not supposed to for safety reasons. It just depends on the person in costume.
They also know that some of the kids are afraid of them, and they're trained to deal with this too. Some of them go very easy and let the child make the first move (which is often more successful then you might think), some of them know to leave the child alone (and if not, they usually get the hint soon enough), and...unfortunately, some of them just don't care. Again, it just depends on the CM in the costume.
I remember when I was 4 or 5 years old, during my second trip to
Disneyland, I was approached by Roger Rabbit. Although I was a fan then and was even wearing a Roger Rabbit shirt and hat, I was mortally terrified of the characters at the time and I almost ran screaming from him. But he was trained well and went easy on me, I warmed up a little bit, and it turned into a magical 15-minute experience that I'll never forget. It ended with me being picked up by Roger and given the biggest, most affectionate hug I'd ever received, with the crowd behind me going "AWWWWWWWWW..." in turn.
So yes, don't be surprised if a character comes up to hug you or your kids. It's what they're trained to do, and most of them are very good at it.