Help with my dog and crate training?

I recently adopted a 1 year old poodle/shih tzu mix from an animal rescue. He is the sweetes most loveable dog I have ever met. However, he has two fatal flaws -- bathroom time and crate training! My dog refuses to poop outside, even after 30 minutes or so of walking and walking in the gr. So of course when I get home from work and open the crate, my dog is a disgusting poop-covered mess and in need of a bath. I am losing my patience with him fast. I know it takes time to train but I thought dogs NEVER went potty in their crate...the whole purpose of the crate in the first place. I already wake up at 4:30 in the morning to take care of this dog. I cannot get up any earlier. I don't want to return him to the shelter but if this doesn't stop he will go back. Can ANYBODY tell me what I'm doing wrong? He gets praise and treats when he goes outside. I take him out MANY times a day. He still refuses. It's almost as if he LIKES to lay in his own crap.