Approximately how often do sheep produce wool? I set mine up with a doghouse and fed it, but haven't seen it produce yet today

I honestly don't know. Usually I feed mine to full and by the time he's down to 50% or so, he's prepared a nice number of wool. I think it's hunger point based.
Keep him fed and make sure he's attached to a doghouse and he will produce wool for ya.

Something perhaps worth noting: it's possible to become a tamer and not have a pet, but level up by petting other people's pets. I've gotten the Freeganism skill from Scavenging and have also been feeding others' pets with the free feed I scavenge. I can't clean up their droppings or anything, but I've reached level 2 without having a pet of my own (mostly due to petting them - one skill point per different pet).

While you can level up tamer without a pet, you still need a pet to use the pet specific abilities like Circle of Life and Treasure Hunter. This was more of a brief guide on what tamer abilities exist and how to get started with taming. Here's a guide that talks about actually leveling the tamer ability.

Valentines Shark Puppies also hate bird food. I fed mine some and it didn't gain any hunger points.