e46 318 2003 AC problems!

My cooling system is literary driving me crazy. A few weeks ago the AC went off completely. When I turn it on the fan just blows in hot or warm air depending on the state of the engine or outside air.

Now my cooling system has decided to get a mind of its own. The temp sensor keeps flipping from -40 degrees Celsius to +50 degrees! I would check the coolant level and sometimes it would be low and I would top it up with coolant. But many times it would be ok. Now the coolant warning indicator keeps coming and going off when I am driving.

Sometimes I feel the engine hot from the side where I rest my left foot (car is RHD) while driving but sometimes it becomes very cold. At other times when the coolant indicator comes on, I feel some vibration when I gently try to accelerate!

What is happening? Is there a leak in the cooling system? Why does the coolant keep getting low? Do I need to replace the whole cooling system? The Odometer reading is 85k now.

I would like to have an idea what is wrong before I take the car to the mechanic to check my whole system - AC and engine cooling system as well! Any help will be very appreciated.

Thanks