No comment on what is going on with the trespassing. However, I am telling you that leaving a chained puppy or dog in the backyard while you are gone is risky business. A friend of mine at work found that out the hard way. He left his puppy yellow lab chained while at work. He came home to find the puppy had twisted itself all up in the chain and the puppy was dead. It's just not a good idea, man. You may want to secure that backyard and get rid of that chain.
Out of curiosity, if I installed some sort of trap and the trespasser falls into it, killing him, am I, in any way, liable?
Of course you are. That's what insurance is for. Any guest whether invited or not.... if they are injured on your property. You're responsible for.
Do yo have a kid that goes to that school you say is nearby? or a nephew/niece, Could be them skipping school and hanging out in your backyard smoking weed and stuff with their friends. Also might explain why they haven't stolen your puppy.
pull a "duck season! rabbit season! rabbit season! duck season!" on them and leave the door open and see if they'll close it for you.
I would just wait out for them, catch them in the act. If they were kids, talk to their parents. My brother caught some teenage kids thrashing my mom's plants around in our backyard, tearing out the roots, and he chased down in his car and threatened them to not even set foot on our street again.
Since it happens frequently I would suggest it is a neighbor and would keep an eye out for them. This is indeed weird behavior and be careful man. Stay home one weekday and find out who it is.
Intentionally harming anyone is against the law outside of a very few exceptions. Insurance will not protect you from that.