I'm a veterinarian in the US, and I definitely recommend this dog have blood work done as well as a urinalysis by cystocentesis, so the vet can check for masses or stones in the bladder. Bassets are
famous for getting stones in their bladder, which cause a lot of discomfort and UTIs. Bladder stones can occur even in quite young dogs. A healthy dog should not be having accidents in the house, and it should not be related to his intact status. However there are several fairly serious health conditions that he needs to be screened for which could be resulting in inappropriate urination in the house.
You are correct the abdominal or inguinal testicle does need to be removed, to prevent it becoming cancerous. It's your choice whether they remove the other one. There are health benefits to having him remain partially intact, specifically protective effect against some other aggressive cancers, but you're going to have to push back on your vet if you want him or her to perform only a partial neuter.
As far as preventing him from peeing in that area, I always recommend what we call an x-pen in the States to my clients, where he can be contained when you're not with him,
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don't know if you have another name for it. These are available online or at any large pet store. A dog who is having trouble staying clean in the house should be constrained to an area with vinyl flooring or that's easily cleaned when you're not watching him, either with a crate or x-pen, until you have identified the cause.
You will need to do everything you can to remove the odor in that area as well, as part of retraining him not to mark there. Try several enzyme-based Digestive cleaners, and invest in an inexpensive blacklight flashlight so that you can find all of the remaining dried urine. When the area no longer fluoresces you will have gotten rid of it.
Cheers