I've been advised against leaving paws wet and not blow drying, for fear of undermining the health of the various joints in there. It sounds a bit far-fetched to me, especially if the dog will just lounge about in a warm house though.
Re the underside, it's not just the underside that needs cleaning; it's also the draping side hair that is stained as well.
It does seem that the only option is to just rinse/wash the underside in the sink. I am just worried because vets here advice washing the dog at the most every 25 days. Probably to avoid removing all the oils from the skin?
That's silly to only wash a dog every 25 days! I sure wouldn't want to snuggle next to Kodi on the 24th day!
Kodi gets washed about once a week and has since he was a tiny puppy. I wash myself daily, and _I_ don't have a problem with my skin! They are no different. Use quality shampoo and a good, moisturizing conditioner and their skin will be MUCH better than that of a dirty dog.
There is a GRAIN of truth in the idea of bathing less frequently, but it is for hunting breeds, not house dogs. Hunting breeds develop a protective, oily coat which keeps them dry and warm in the field, or even swimming. Our Havanese don't have that kind of coat. They never develop that kind of protective, oily outer coat whether we bathe them or not. They are basically little sponges! (as you have, no doubt, already experienced!) So unless you are planning to train Mucho to retrieve ducks, it's OK to bathe him as often as you want.
As far as leaving their feet wet after bating? Where we live, if I had to dry Kodi's feet every time he went out either in the rain or in the dewy grass, I would do NOTHING but dry his feet all day every day. I have never heard from anyone that it can have any bad effect on their joints or feet, and Kodi is a performance dog. he works hard on a daily basis.
When Kodi comes in with dewy feet or wet feet from rain, I simply "towel dry" them with a micro fiber or "Shamwow" cloth that is very absorbent. Then he air dries the rest of the way. If he is out on the street or in the woods and actually gets muddy, I either rinse off his belly and legs or wash them with a bit of shampoo and conditioner in the sink, depending on how bad they are. Then, again, I towel him dry and let him air dry the rest of the way. I often also put a big beach towel on the couch for him if he's damp. He likes to rub up and down all over the towel to help dry himself more.
As far as the pee on his undercarriage… Not sure what's going on there. I have heard of this with some other dogs, but Kodi never gets himself dirty or wet (with pee) under there.