Are you sure about the compressor not engaging when you turn on AC?
If you are then you likely have two issues maybe three.
Your issue with the control of the heat and possibly the AC not blowing cold is the blend door. It has likely failed. Did you have any foam blow out of your vents? That's the seal on the blend door blowing out.
Your issue with the control of the air flow is the mechanical switching attached to that dial, it likely broke inside the dash somewhere so when you turn the dial it actually doesn't turn anything.
If you clear up both those problems and still have no AC then you've also got a problem with your AC system, which may be as simple as more coolant being needed.
If you are then you likely have two issues maybe three.
Your issue with the control of the heat and possibly the AC not blowing cold is the blend door. It has likely failed. Did you have any foam blow out of your vents? That's the seal on the blend door blowing out.
Your issue with the control of the air flow is the mechanical switching attached to that dial, it likely broke inside the dash somewhere so when you turn the dial it actually doesn't turn anything.
If you clear up both those problems and still have no AC then you've also got a problem with your AC system, which may be as simple as more coolant being needed.