Engine overheating and transmission issues

I bought a fairly used 2011 Toyota corolla (as is) about 3 weeks ago. I made all the changes I could identify after purchase. But there is still 2 pertinent issues.

1.) I have an issue with my engine overheating. This only happen when I use the AC on a hot day. The radiator fan is working fine.

2.) My transmission stick does not lock the gears properly when I initially put it either on the D, N or R positions unless I shift the stick out of those positions slightly, then return it. Then it locks into the appropriate gear.
3.) Sometimes when the stick is on the Park (P) position, the car will not start until I shake the stick side to side. Then the car will start.

1.) I live in Nigeria-(West Africa) The weather here is very very hot everyday. Does the outside hot air temperature also affects the coolant temperature in the car?

The coolant temp reading starts rising and approaches the "H" level on a hot day when I am using the A/C. It rises faster when I am in a traffic jam or slow moving tracfic on a hot day.
To get the reading down on a hot day, I have to swicth off the A/C and the level gradually goes down to "slightly above" the middle mark and it stays there.

Also, when the A/C is ON and my car is idle (not moving) and my transmission stick is on D position (my feet is on the brake because I am in a traffic jam) my engine vibrates/shakes and the Tachometer needle begins to fluctuate between 0 and 1 intermittently.
It gets worse as the coolant temperature increases to the "H" level. Then suddenly the Tachometer needle goes down to the 0 position and my car just switches off right on the road. I have to put the gear stick on Park position, turn off the A/C and restart the car again to move. Because I am in a traffic jam or slow moving traffic this happens intermittently. I have to restart my car several times on the road because of this. It does not happen when my car is moving freely or when the A/C is off. Also, the shaking and the Tachometer fluctuations does not occur when my gear stick is either on the N Position or R position

So this issue only occurs in slow moving traffic, when my car is idle, when the A/C is on and when the gear stick is on the drive position "D"

Then I begin to perceive a burning rubber smell right inside my car

I want to point out that the muffler was removed before I bought the car. I plan to replace it soon. If the absent muffler is part of the reason why my engine is acting this way please let me know.

Can some one please help me out here. I just don't know what the mechanics have done about this but the situation is not resolved and I am sure they don't really know where the fault is. If I could know then I would simply direct them on what to exactly fix in my car instead of them doing trial and error.

Is my radiator and transmission bad?

Apart from this when I am driving in the evening with my windows down and when there is no Sun at all and I am not using the A/C, the car runs great and the engines responds very well. in fact its the only time period in any day that I actually enjoy driving this car. I just want the same experience for day time/hot day driving.

Your feed back is highly appreciated
1. You need air flow across the radiator in order to cool the engine. Stop and go driving in extremely hot/dusty conditions just adds to the problem. I would suggest having your cooling system checked. There are so many factors that could contribute to your problem (thermostat, water pump, radiator cap, and radiator). Are you using Toyota log life coolant (pink) or another type? Some coolant brands require it to be diluted 50/50, while others come pre-mixed. Also your radiator cooling fins could be clogged internally with segments (hard water can leave mineral deposits) or externally with debris (bugs, road trash, etc.). Also check the radiator and water pump for leaks. If you find no leaks, I would suggest replacing the coolant first. I live in Arizona where summer temps exceed 100+F and I haven't had any cooling problems.

The idle typically fluctuates when the AC is on, but not enough to cause the engine to stall. I suggest cleaning the carbon buildup around the butterfly in the throttle body. You can find "how to" video's on YouTube. If your car has drive by wire, be sure to find a video specific for that. Yes, removing the muffler may have a negative performance impact. The ECU may have a hard time adjusting air/fuel metering at idle (provided your vehicle is equipped for emissions).

2. Sounds like something is loose either with the shifter itself or shifter linkage. If you take the shift console off, there may be a clue hiding in there.
Limit your a/c use in traffic. If you don’t have to use it don’t. Plan your trip instead of sitting in traffic. Heat is number one killer of almost anything. Arizona is very hot but the traffic is always moving. I was in Asia for business and the weather was 100 degree and the traffic was bad. A 15 mike trip was 2 hours. I was in a new rental car and the car was getting hot I had to turn the a/c off and on in traffic. The needle was in half..

As far as your transmission have them check your linkage. It sound like a minor problem or hopefully.
Hi there,
Concerning your heat up problem i would suggest you to check your car after having run for some distance while engine still running when you notice temperature has risen excessively. But be careful, you might get surprised to notice a major leak in radiator hose as pressure is great. i could notice the jets caused by leak in rubber. May be first thing to do is to check level of coolant and notice if it goes down after driving some kilometres. I topped up missing coolant and notice the leak after some time when is got worst and rubber hose nearly half open.
Check for water pump if working correctly you will notice any leak if losing water.
It's one of the most reliable cars that I've ever owned. I had to make at the time and just picked the first cheap buy that I could find on Craig's list. Also Toyotas have great history of great gas mileage. So that played a big part in my search for a car. The 1988 model had a stick shift and it's 5 speed.