There were two television programmes that explored what would happen if there were no people in the immediate future. One of those is the growing number of feral and stray dogs which is probably already happening in deserted places elsewhere. That and being taken over by wildlife in general. I’ve seen abandoned buildings give way to those.
Unless if somebody turns it into something else. Also while robots and pets can help alleviate loneliness for some, they can’t entirely offset population decline. Immigrants on the other hand help offset it but could also impact the cultural and ethnic makeup a lot especially if they exert enough power and influence.
On the subject of pets, nobody wants to deal with inbred pets but to limit it would mean being more open to interbreeding and necessarily less restrictions on breeding in general especially with stray dogs but that also leads to another problem. Any solution to and effect of declining human populations aren’t without their own problems but they’re inevitable.