would a surge protector with an appliance that might be shorting, protect my house circuit?
I believe you have something else, and that something else is SERIOUS, that has nothing to do with your surge protector.

Yes, a surge protector can caused a short circuit if the active portion of it, in almost all cases a device called a MOV or Metal Oxide Varistor.  A MOV is designed to do nothing when the voltage across it is "normal" but to conduct electricity when the voltage is too high.  The problem is that the MOV literally sacrifices itself to protect whatever is plugged into it.  Usually there is a circuit breaker on the surge protector that is designed to trip when the MOV activates.

Over time the MOV wears out from this "abuse" and it may develop a permanent short which renders the entire surge protector useless and it should be replaced.  Note that is may also fail "silently" where the MOV no longer conducts at high voltage and the entire surge suppressor doesn't do what you expect it to do.

This does NOT appear to be what's going on here, however. Even if it were totally shorted out, which is not what you're describing, it would only trip the circuit breaker that it's plugged into, not the entire apartment.  (That is assuming your apt. has more than one circuit in it.)

It sounds like you have a more general wiring problem and you should get your landlord to hire an electrician to investigate it before more of your electronic devices get damaged or someone get injured.