To me this definitely sounds like a faulty thermostat. The thermostat is supposed to gradiently open as the temperature (and pressure) of the coolant approaches a certain range. It is entirely possible that there is buildup or other faults causing it to stick, until the pressure builds enough to force (pop) it open, then causing it to cool back down to proper temp.
An easy way to check this (though not as accurate as an IR thermometer) is to start the car, and after a minute or 2 check to see if the upper radiator hose is hot/warm by carefully touching it with your hand. If it is not, you can then also check the bottom hose (which you may have better access to from underneath the car). If there is a significant difference in temp between the 2 hoses, then it is likely that the tstat is not opening as it should. USE CAUTION! Any time the engine is running you must be very careful poking around in the engine compartment. There are lots of fast moving and very hot parts. Don't go poking around unless you're totally comfortable.
It is critical to get the overheating issue resolved as soon as possible. Even just being in the red briefly is really hard on the motor and repeating this often is going to cause more damage and can total the engine given enough repitition.
As for the AC issue, it could be due to low charge or possibly something faulting the coil/clutch function. Honestly there are many possibilities with this. AC is also susceptible to issues caused by the ambient temp being too high, so it is possible that just fixing the engine temp will fix the AC as well. If that, or just recharging the gas doesn't fix it, then you'd be best served getting the AC system diagnosed. Just know that it is much easier to diagnose if everything else on the vehicle is working correctly.
My best advice would be :
- Check the hose temps,
- It is possible that air bubbles are getting stuck at the tstat, and preventing it from opening correctly (the air not being as hot or having as much pressure as the coolant), so you can try bleeding the coolant to get any possible air out (here is a pretty good instruct) and some systems are more susceptible to the air issues than others,
- Replace weak, worn out, or leaking hoses,
- If you did have bubbles, consider replacing you radiator cap at this point,
- Replace thermostat,
- Flush cooling system (in case there is contamination loitering in there - you may be able to see mineral buildup around the valve or spring parts of the tstat or debris in the coolant when it's drained),
- Top of coolant levels with new, properly diluted (and correct type) coolant,
- Make sure to bleed any bubbles out again,
- Rinse off radiators and gently brush any stuck bugs or debris (be careful not to use much pressure with the water or brush because you'll bend the fins and reduce efficiency and use a fine plastic bristle brush, not metal because this will scratch or damage the fins and lines),
- Make sure all belts, pulleys, and fans are properly moving,
- Add more refrigerant if needed.
Beyond those actions you may start getting into something bigger than you want to tackle, so a shop may be the best option after that.