Jenny, on the dog swimming idea. What kind of dog is it? I've forgotten. Some of the critters love the water. Our lab/hound puppy is jumping in any puddle she sees, but our border mutt wants nothing to do with wet and will only get in if we're in the creek or something and he feels compelled to come along. It's rarely his idea, though.
I agree at least one walk a day is ideal, but I find that sometimes that doesn't happen around here, either. We also have a big yard (and are getting a bigger fence). My problem is DH works a bit late (doesn't get home til 7 often) and sometimes my kids just don't want to walk the dogs and they can't really keep up with a faster pace anyway. On the days when they're both in school I can usually get a walk in and on the weekends we can usually swing it, but on the days when dd2 is home with me it just doesn't always happen. Luckily, young pup plays a mean game of fetch and actually gets much more physical exercise doing that than she would on a walk. I don't think it really subs for a walk because they need to get out and see the sights and smell the smells just like we do. I know I don't want to be in the house and yard for weeks on end, but playing in the yard does do a good job of exercising her when a walk just isn't in the cards.
On the pooping, can you get a baby gate across the bathroom so you can keep an eye on the pup's progress, but still keep the dogfood out of toddler's reach? Or could you just occupy the toddler for 10 minutes? A lot of dogs do better on a feeding schedule. Ours are on a very loose one and we do free feed which is not usually recommended, but it's worked well for us so far so I didn't want to upset my older dog's apple cart by doing something new. Most folks recommend putting the dog dishes down once or twice a day for 10 minutes and then taking them up. The dog will learn to eat at that time pretty quickly. Then you immediately take the dog out. Most dogs, and especially puppies, do need to poop immediately after a meal. Establishing a regular routine is the key to getting over this pooping in the house problem. I also agree that feeding in the crate could be a good way to go, but again I would try it for 10 minutes twice a day and then put the food away. I mean, I would, but I don't! If I couldn't get the housebreaking thing under control, though, I would.
P.S. A Premium quality kibble is probably made with better ingredients than a lot of human food. I found when we switched to a cheaper kibble our older dog's coat lost a lot of its luster and shine. DH wanted to go cheaper, but I buy the dogfood so I switched to a different premium (not super-duper premium, but good quality) kibble and that seems to have improved matters. Young pup looks super and our older guy has regained a lot of his shine. Not that I want to feed it to my kids, but...
P.P.S. Here's an article that might be helpful on pooping and peeing in the crate:
http://www.mysmartpuppy.com/articles/Goesincrate.htm . I saw his picture from your other post and see he's a chihuahua or chi mix. I think they have a reputation for sometimes being difficult to housetrain. I'm not a chihuahua expert, though.
P.P.P.S. Just wanted to add that you should clean up every poop and pee accident with an enzyme cleaner like Bac-Out or Nature's Miracle to get out the smells that we humans can't smell, but the doggy noses can. When my pup was pooping on the deck this winter (I had some sympathy — it was cold, she's shorthaired, and it's a long flight of stairs out to the backyard) I also shoveled her poop into the area that I wanted her to go so that she would know the right spot. She caught on. (Also the weather warmed up and she got bigger.) She hasn't pooped on the deck in a long, long time. Good luck!