I feel for you bro. I've been there, but not nearly as bad as your circumstances.

I bought a home and did not know about the tiny little dog behind my property.

I have a dog.

The problem is, people leave there dogs outside all day because they clearly don't want it anymore.

My dog goes outside to poop and pee and comes back in to hang out as a family member.

The dog behind me, is a yappy small dog, which they leave out because they seem to forget they let it out in the first place. It almost immediately starts barking at nothing until owner tries to call it in.

Put thier animal to sleep in a noble way. They obviously do not want it if they ignore it and leave it outside for any more reason then it has to.