This is not a problem I am confident of solving. It is very difficult to train a dog not to do something when you aren't around. This sounds to me like a case of marking, urinating to show you are top dog. Changing an established habit at his age will be hard. It may have to do with genetics, late neutering, and previous poor training. He may have been dumped at the shelter because his last owner couldn't solve the problem. It is possible a good course of obedience training would help. The dogs see all the people and dogs in the household as a pack with each having their own rank in the pack and a top dog. Life is much easier if the 2 legged pack members outrank the 4 legged ones. You can learn to play the role of top dog by reading some books or going to a good obedience class. A good obedience class or book is about you being top dog, not about rewarding standard commands with a treat. Start at http://www.dogsbestfriend.com/
Likely the best solution is to limit him to a very small area when you aren't around, a tie down or crate. Some older dogs adjust better to a tie down, a short length of chain secured at one end and a snap at the other. Other dogs may not be as bad as the young Labs I am plagued with. Still your house and dog will be much safer with the dog in a crate when you are away. The dog may be happier in its den than loose in the house. It relaxes, it feels safe in its den. It rests, the body slows down reducing the need for water and relieving its self. Dogs that have been crated all along do very well. Many of them will rest in their crates even when the door is open. I think the plastic ones give the dog more of a safe, enclosed den feeling. They are harder for dogs to open too. Metal ones can be put in a corner or covered with something the dog can't pull in and chew. Select a crate just big enough for the full grown dog to stretch out in.
Leave it some toys. Perhaps a Kong filled with peanut butter. Don't leave
Anything in the crate the dog might chew up. It will do fine without even any
Bedding. You will come home to a safe dog and a house you can enjoy.
A dog that has not been crated since it was little, make take some work.
Start out just putting its toys and treats in the crate. Praise it for going
In. If you have been able to trust it with any bedding, put that in the crate.
Feed it in the crate. This is also an easy way to maintain order at feeding
Time for more than one dog.