She was upset all locks were changed, and filed a police report that there were people trespassing in her property.

If I place my own "no loitering" sign, will the police honor this and fine them?

If I were in your shoes, I'd simply explain to the downstairs tenants that when the noise is going on, they should call the police and not you.

After I got home, the tenant called and apologized and agreed to leave the apartment rather than have me file a police report.  

I'll contact you later" type of response).Trouble for me is, I live overseas, so physically visiting this person is a bit difficult, but I'm not adverse to the idea, for the principle of the matter.My question is, at what point do I call the police? 

I would imagine that ANY city would prefer to have a house like that demolished to polish up the area. 

Linoleum actually is all natural and buffs and polishes beautifully.  

I'm a police officer so I totally understand how war zones could be risky but not all people are bad I believe. 

Learn a LOT about investing out of state.Research the market you are going into and figure out the best markets.Make connections in those markets and vet those connections.Save up more money and move up from the war zone and go to a $30,000 to $50,000 minimum market where there is less crime, less vacant lots and vacant houses, where the police will actually respond to a call from a homeowner, where there are jobs for your tenants to work at, where there are schools for your tenants to send their children, where there are real grocery stores for them to buy real food (not gas station or fast food).  

Tell the other tenants to call the police regarding the noise from the karaoke.