Is it my A/C Compressor? 2010 Ford Focus SES?

So I've been taking my car to various mechanics to check my car because it's been acting up. My A/C went out earlier this year because a mechanic left the Freon cap off. Recharged it and it worked till I stopped using my A/C a few months ago (not sure about now since I haven't used it). It sounded like my serpentine belt needed checking. It also went from sounding like a normal car to a small truck. I took it to a mechanic around Thanksgiving because my car stalled and had a good deal of white smoke coming out. They said nothing was wrong. I took it to get checked again yesterday because the white smoke happened again with a different smell (along with problems braking) though it didn't stall. They said nothing was wrong and I went home. Last night the white smoke came back, braking issues till I parked it, and it stalled.

Today took it back and they say the compressor is locked up and they need to replace it with pulleys and belts. $1000-1500. Should I take it to get checked somewhere else? I typically trust these guys, but I'm frustrated.

Get your own a/c compressor/belt off of rockauto.com (good prices), Their trying to rip you off.
Your old compressor didn't lock up because they left a freon cap off, the compressor has a"low freon cut out switch" on it and it will save your compressor, not make it go out. Its more like the bearings in the compressor went out.

If nothing else, take it to them with your own parts and have them change it out, 2.5 hrs labor

Fix the AC next summer. Engine off, turn off AC if it is not off. Open the hood. Have someone start the car while you listen to the engine. A locked compressor pulley will make the serpentine belt squeal really loud. No squealing, turn ON the AC. A locked compressor will make the belt squeal and likely reduce the engine rpm or stall the engine.

Your first concern is the white smoke. Go to a different garage and have them check if the head gasket is blown with a chemical analyzer. You can check the oil and/or coolant if they are milky but this is not a true test.

That cap does nothing but protect the gas valve from dust. Leaving it off did not cause the ac to stop working. If the gas leaked out the leak needs to be repaired. A locked up compressor needs to be replaced. It likely burned the belt as its pulley dragged the belt. If all the gas and lubrication leaked out of the compressor, that's why it failed. The brake problem… Have a mechanic to examine the brakes. Car sounding like a truck… Perhaps the exhaust system leaks or is loose. That car sure is falling apart to only be a 2010. Regular maintenance might have prevented most of that. It isn't the mechanic's fault there's bad news about your car. Stuff like brakes wears out predictably.