Um.. well like I said I took the car to a AC Recharge shop after having the NEW AC Compressor and AC Condenser installed into the car, the Heater Core was also installed at this time.

The shop was the one who pointed out the hose in the picture.

I'm trying my best to correctly answer the questions you ask.. so bare with me..

I can hear the fan run when engine start and goes off of course.. if that's what you mean, the air does blow from all vents.. its just blowing cold air (but not cold... it feels like as if a fan is just blowing in a house.)

I wouldn't know if the compressor runs?? how would I find this out anyway, the only thing I know is that it was a new compressor, condenser & heater core installed ??