What do the police and landlord suggest you do? Have you tried their suggestions? What happened?
I suggest that either you have to be brave and consistently tell them no in a firm way or you have to move. When you make nice and go along with themyou are telling them what ththey're doing is ok. Other than slashing tires once have they damaged property or injured anyone? I suggest that they may be controlling you thru intimidation.
Have you tried laughing at the absurd? I wonder if this would lessen if you responded in a normal way? Be pleasant but be real. Say something like "seriously ? You want a threesome? You'll have to find someone else" and laugh as you walk away. Do not show fear. This is so crazy. Try not taking them so seriously.
If they haven't made angry verbal threats and only slashed tires I would try not feeding into their fantasy. I'd run this idea by the policeman working your neighborhood.
In Portland we have a neighborhood livibility office. Also a neighborhood mediation office. If your city has something like that they may be able to help you.
I was a police officer for 20 years and I've never heard of anyone having this sort of neighbor. Neighbors who are angry, make threats, fight, put garbage on the street, are unfriendly etc. Or neighbors who are paranoid and mentally ill in a less obvious way. I wonder if you're not understanding the dynamics of this family. They sound like huge jokers who get a kick out of scaring you. Yes, this is juvenile but some people never grow up.
Do NOT pour cooking oil on your steps or do anything else to injure them. Perhaps that suggestion was made in jest but wording sounded serious. They can sue. Then your problem will multiply.
After your SWH: I'M saying there is a middle ground between "sugary sweet" and moving. Try the direct approach. "Hey, this is no longer funny. Stop with the jokes" and walk away.
This message by its self does not sound creepy. A helpful response from you would be "Thanks for your offer. However, I'm a private sort of person and I don't want to be a part of your family. I think you mean well and I'd appreciate you not trying to be family."
Because of the way you describe yourself as being sweet and not wanting to upset them I suggest that your issue is one of boundaries and how to inforce them. I get that you're afraid. I really do. I need to say that being this way sets you up for this sort of behaviour from social idiots, especially ones who enjoy making others uncomfortable.