engine runs like a piece of poo when air is on

I'm suspicious that your A/C compressor is binding, the expansion valve is faulty, or maybe is overcharged, more likely binding or a faulty expansion valve. It's also possible that the engine cooling fan is not working right, causing the A/C compressor to heat up and that could damage it. It's normal for the A/C to load down the engine, but the car should still be functionable. At idle, the idle air control (IAC) in the throttle body should compensate for the additional load but although it would be noticeable, it still should run well. At speed, it would also be a little noticeable but not an annoyance. I think there may be a problem waiting to get worse and cost more. I recommend you get it checked by a Mazda or A/C specialist ASAP, I know it's hot in NC so you don't want a breakdown which always costs more. If you want to find a Mazda specialist, here's our directory link for you: https://repairpal.com/mazda-repair-in-burlington-north-carolina As far as the miss, You need to find out why that's happening. You don't mention the check engine light so I guess the miss isn't bad enough to generate a diagnostic code, but if the light's on, it's trying to tell you something important. FYI: The check engine light indicates that the computer has detected a malfunction in the engine/transmission control system and has diagnostic codes and data stored which would be essential in diagnosing the problem. A technician with a scanner can access the diagnostic codes and data and then be able to give you a diagnosis.