That's one of the reasons it's not a good idea to purchase a dog from the pet store... They usually have the puppies there all day peeing and pooping in their kennels that they lose that instinct not to soil their surroundings. Makes it really difficult to potty train them babies.
But yes, it is not too good for a dog to lay in his own pee. It would be better to put him in some kind of a play-pen type thing where one side is his crate, then a pee pad or something on the other side. This is not ideal either - the dog learns that it's okay to pee inside the house... But at least it is better than rolling in his own pee. A dog's pee has a lot more meaning than just pee. It carries psychological signals through pheromones that is part of a dog's communication. A dog rolling in his own pee has a psychological impact on top of the unsanitary health concern, so a playpen would help.
As far as daily shampoos - they have "dry clean" shampoos. It's a powder that you apply then brush off. It can cause skin allergies though for those dogs sensitive to it. Also, I don't really recommend this for pee'd on dogs because it doesn't dis-infect the pee - it bonds to oils in the dog's hair that then lifts up the dirt that you can then brush off with a dog brush and then it deodorizes - or a more appropriate term, masks the odor like perfume would. The pee needs to be dis-infected. But, I use it for my bichon for touch-ups in between his salon appointments.
Okay, for a mild shampoo, I make my own (I had it for my bichon but it works for bullys too). The shampoos in the store are designed to be marketable - so it has things like "sudsy" and "nice smell" so people will buy it. These are added harsh chemicals designed for the people and are not really necessary to clean a dog. So, here's what I use:
22 oz. Ivory Liquid Dish Detergent (or any mild, hypoallergenic dish detergent)
2 oz. glycerine (you can buy this at the pharmacy in small 4 oz bottles. I like the pure vegetable glycerine ones. I buy mine by the gallon at the tack and feed store where I buy my dog food)
2 cups plain white vinegar
1 gallon water
That's it. It's not sudsy, so don't expect all those luxurious bubbles. But, it cleans very well and it has a lot less chemicals than your store-bought dog shampoo. And it's cheap too.
Hope this helps.