I think your best bet is a mandatory quiet time for the toddler - if she won't sit on her bed, see if you can get a little cot or one of those foam chairs that looks like a mini-version of a standard living room chair. My stepdaughter had one for each kid, and the bottom cushion lifted up and folded out, creating a sort of chaise lounge. The kids would crawl in their with a book, a pillow and a little afghan, and half the time they would doze off. It was easier than "enforcing" a nap when they are resisting it. Tell her she doesn't have to sleep, but she needs to rest her body and have book time. You can also put on a calming CD if you have a small player, but have it play endlessly so she isn't getting up to change it. If she naps, great. If not, she'll at least recharge a little and not be yawning at dinnertime. If you can put her to bed a little later, sometimes that works. Is her room super dark? The rising sun sometimes signals "play time!" to them. You could also get a white noise machine (we use one of those allergy air filters - not sure it does much to soak up house dust but it was on sale and it makes a constant sound to drown out the early morning traffic like the paper delivery guy and the two neighbors who are out jogging and chatting super early.
How old is the puppy? Where is the puppy sleeping? Too close to her room? Can you change that? Puppies have to go out in the middle of the night - that's just how it is. Same as a baby getting up to feed and be changed. Take the puppy out at the last possible moment (11 PM or whenever, and see if you can keep it in a dark area or put a blanket over its crate. If they hear the birds chirping or sense the sunrise, it's all over. But there's no way he should be running around and waking everyone else up. You or your husband need to take the pup out right away (yeah, I know - no one's moving fast at 4:30 AM except the pup!) but you've got to get him out and quiet before the kids wake up. I'd try to put him back in the crate with something safe to occupy him - a Kong stuffed with peanut butter or some sort of quiet chew toy that's safe when someone's not watching. Ask the pet store or vet. If the dog is sleeping with you, you don't want a toy that involves a lot of licking or chewing, as that will keep you from going back to sleep too!