I shorted a circuit, reset a breaker, now much of the house electricity only works if I turn on a wall AC unit

The load center with the transfer switch (or "critical load panel") should have a circuit for each circuit that was specified to be connected to the generator. The original breakers no longer function because nothing is connected to them now; there is probably one dual-pole breaker feeding the critical load panel, and there should then be a breaker for each circuit there.

If you are not comfortable investigating, call the electrician that did the install and ask them to walk you through operation and service requirements for the setup.

Probably a bad breaker, but the symptoms are also consistent with a mild overload causing the breaker to heat up and trip over time, the longer you let the breaker cool before resetting, the longer it takes for it to heat up and trip again.

I'd leave it turned off until you've ruled out a short in the wiring or other overload condition causing it to trip - if it's a short somewhere, that short is causing a lot of heat somewhere in the walls or other inaccessible place.

It's possible that the wiring is not straightforward and there's a load connected to the breaker that you're not aware of - like maybe the toaster in the kitchen is on the same circuit so be aware of what other appliances in the house may be in use when you experience the trip.

The hard thing about finding a short circuit is that if it's due to, say, a bad connection in a junction box under the floor, it may not short out until you walk near it and make the wires flex enough to touch. If you notice any kind of burning smell associated with the trip, keep the breaker turned off until someone finds the problem.

Some types of breakers are subject to "nuisance tripping" even without any overload or wiring problems - if you have an AFCI or GFCI breaker, you could be experiencing nuisance trips due to a device that's plugged in.

If you have 20A service to your bedroom, you're probably in a newer house (15A is more typical, especially in older construction), so a short is less likely than if you're in an older house so the most likely cause of the problem is a bad breaker. Hopefully the landlord had his electrician swap out the breaker since a breaker is relatively cheap compared to having the electrician make another service call.

The consequences of a short somewhere are severe enough (i.e. a fire in the walls or other inaccessible area) that it's worth ruling out a short before you leave the circuit turned on. If the landlord can't fix it to your satisfaction, hire an electrician of your own and bill it back to the landlord. Even if it's just a bad breaker, the landlord is responsible for wiring problems.

The fact that it stays on after waiting 5-6 hours probably means that it's not a short, but I wouldn't dismiss any repeated circuit breaker trips since the circuit breaker is your only warning about a dangerous short in the wiring.