I want to know if there is a problem with my 2006 audi a4 engine

you have left a wiring loom connection off from the sound of it ,no idea where but have a good look tomorrow,best of british with it

May be a vacuum leak making it idle rough and higher rpms than normal . Did it suddenly do this ? Any recent work done on engine ?

It can be a problem that is simple to complex. Best see a mechanic for a personal inspection and recommendation.

Checking the pressure with the system not working won't tell you anything. A system with just a half pound of freon will read outside air temp on both gauges with the compressor off.
I guess my first few questions would be - - is the compressor running at idle, what are the gauge readings at idle, and is the cooling fan running at idle ?
The two gauge readings at idle with the compressor running should be 30-45 low and 200-250 high.

these vehicles are known for ignition coil failure. typical symptoms include: Misfire, power loss, vibrating engine, poor fuel consumption. If the vehicle's engine vibrates, it be misfiring. Is your chk engine light on???

hey not sure if its totally right but the high pitch squeal at when its idleing could be the tentioner pulley bearings on the engine belt not sure about the whining noise when its shut off

Could be a broken motor mount, so that one of the pulleys is scraping a shroud or some other fixed item.
Examine the area around the pulleys and see if there is bare metal below one of them.
If so, have the motor mounts checked.

to get trouble codes you must hook up a test tool called a scanner, this interfaces with the computer in the car and pulls the codes from the computers memory.

Could be a transmission issue. Did you hit any objects while driving ? Have you performed a visual inspection under your vehicle ? Have you checked to see if you have any loose vacuum hoses ? Any codes in your engine control module or transmission control module ?

I had a high pitch noise, didn't sound like it came from the "underside". Turns out it was one of fans that had little chips in the blades? My Audi dealer replaced it, and it's fine now.