Air vents don't move on my alto

does it have the side air vents? 1997-2002.... no air vents 2002-2006

That car has 'blend door motors' that will switch the air flow. You are probably looking a problem with the control head or 1 of the motors that switch the vents and air flow.

had similar problem with my ford behind the dash there is a little elect. motor and gear box that moves the dampers the gears have broke they are plastic

defrost is the defalt mode look for a damaged vacuum line going ti the control head

Well, I don't know. I would have to see the car.
Generally, as long as air can flow thru the condenser and radiator, the ac system can produce cold air.
As for the interior, you normally pull in outside air and cool or heat it, but you can also recirculate inside air which would close off the outside fresh air intake for the cabin.

When you select A/C or Heater, there is a door (vent air flow change) that open and close when making these selections.

Make sure the compressor clutch is engaging by watching whether it spins with the fan belt moving when the engine is on (don't touch it). If the compressor doesn't have its clutch turn when the engine is on, the a/c may have lost its charge again. Take back to dealer to have it checked out, again.

The air direction is vacum controlled. If you have 4X4 and ESOF ( switch in the cab) then you need to inspect the vacum lines that go to the hubs and possibly the hub seals. "default" position for air flow is the defrost position. If you DON'T have 4X4, and it's a diesel engine, inspect the vacum pump, If it's a gas engine then look at all the vacum lines from the intake manifold for a leak.

Most cars & trucks have a vacuum reserve tank under the hood for this problem, some look like a round ball or a can with vacuum hoses going to it, check and make sure it don't have a hole in it, if the take is ok you could add an extra tank to give more reserve vacuum, or if you don't have a tank you can add one.