I have 2 male yorkies. One is 8 yrs old and the other is 11 months. The 8 yr old only has an accident if he is ill. The 11 month old yorkie has the same behavior as your 2 yr old. He is find for a few weeks then seems to mark when left alone. I share your frustrations. I have made mistakes dealing with this in the past.
Based on what you have written, it seems this behavior started with the new female. It would suggest your 2 yr old male is marking his territory for the new puppy. You should check sites on line to deal with this behavior. I have gone to putting my dog in a confined space when I cannot watch him. I also have leashed and tied him to me around the house. This appears to be working and reducing accidents.
Dr Bob
