No cutting action

A powerful vac motor needs lots of air throughput to keep it cool. Some vac motors have a separate air intake and exhaust for cooling air but the majority rely on the same air that passes through the cleaner and sucks up the ash or dirt. These rely on effective filters to keep damaging ash or dirt out of the motor.

While a blocked filter reduces suction and cuts the flow of cooling air, a displaced filter allows ash or dirt into the motor and if early action isn't taken to remedy the situation by cleaning the motor, the life of the motor will be seriously reduced.

The most effective method of motor cleaning after dismantling is by a jet of compressed air...

You probably notice that it blows through the dash UNLESS you accelerate hard, then it's floor and defrost vents that gets the air.

It's a ~$40 dual action bellow sort of up over the gas pedal that is malfunctioning. You can put a band-aid on it by removing the kick panel over the pedals and then plugging the yellow line going to the bellow.

To replace it (per Volvo) you have to remove most of the dash! That's expensive and a pain in the behind, plus you probably will have rattles in the dash after.
OR you can cut a hole in the air box where the baffle connects and replace it, then patch the hole with the 'lid' you cut out and duct tape it back on.
Nobody will ever see it. :D

sounds like overload/ relay cutting out on compressor
cools for a minute or two then cuts compressor out leaving fan only to work

Either the pressure switch is not shutting off when reaching the desired pressure or the pop off is either the wrong one or it is bad. Some blow off at lower pressures for single stage air compressors while others blow off higher for 2 stage air compressors.

It's either the fuse, heating element, thermal cut-off or thermostat

the unit is blowing cold air then tripping out, is it blowing hot air out the back? im sure you mean exhaust air as in hot air so the reason your unit is tripping out is that its overheating and the pressures in the rerigeration system are to high, so the safety trip cuts the unit off.

replace the motor

all the best

the computer that controls the deflector motor are not responding and opening the door for heat, I'm sure you have climate control,check the temp setting sensor to the computer input first

If there is no freon in the system, the AC will not engage. make sure you have freon in the system

sounds like the unit has a low charge and is cutting out on a low press safety. This is more then likey not something you can fit yourself most of this units are not really serviceable with out cutting the lines open and adding a valve. If the unit is still somewhat new you can try for a warranty repair, if not you are better off to replace it.

Is it spark ignition or a glow coil? Sounds like the ignitor is busted. If it was a roll out switch the fan would stay on trying to cool off the switch.