Then they definitely got your name from the funeral home

Did you sign the check you guys sent? The good news is YOU don’t have an income, they can’t garnish you. LOL! BTW, when you called the lawyer did you call the phone number on the letter, or did you look up the phone number in the phone book or online? Scam artists aren’t above using real lawyers names, but redirecting mail and phone calls.

Very likely

I take it she’s the family member you can’t get hold of. Even if they have a lien on the property you didn’t sign any papers, you aren’t liable and a decent lawyer would know that. Like I said, no legal paperwork, no executor, no problem for you. People can bluff all they want, but without a legal contract you cannot be held liable for any of it.
Re-read the letter you got, does it actually read that it’s a bill, or is it a legal notification that there is a lawsuit being filed? If it’s the later one, then they simply sent it to Pete because he’s an heir in the eyes of the law and they are required to do so.