Chances are, you won't have to burp the coolant system. My point was, there's 2 positions on the cap, the looser position allows you to burp the system. Unless you opened up the cooling system (aside from the cap), you won't have to burp. If the cap is tight but still leaks coolant, it probably needs replacing. In a roundabout way, that may cause driveability problems. Part of the reason for the cap is that it keeps pressure in the cooling system. You'll see on the cap that it's marked "13 lbs" or something like that. That means that it keeps 13 PSI in the cooling system, any more and it releases it into the overflow tank. By keeping pressure in the cooling system, it raises the boiling temperature of the coolant. 2 degrees for every PSI. That means that normal water that would boil at 212F, with 13 PSI, you're looking at 238F. Coolant itself raises that even further. If the pressure gets lost and the coolant is weak, you could have a low boiling temperature in the system. That could mean that you're getting steam pockets which would cause issues with the thermostat or coolant temp sensor which would give you all kinds of problems, it's tied into the computer for fuel mixture.
Long story short, replace the cap and top off the system with straight undiluted coolant. That will help out aging coolant that might be diluted, buying you some time to flush the system properly in the future.
Can you get a picture of the trans cooler lines? If they're pinched, they may restrict flow which could cause issues with the trans, the least of which would be overheating. However, you probably shouldn't get too focused on this issue as it doesn't have much relevance to your drivability problem. That said, it wouldn't take much to cut the kink out and replace it with some rubber hose. A small/local auto parts store may help you with that.
*EDIT*
If you happen to have access to a multimeter, you can check the ECT (engine coolant temperature sensor) and IAT (intake air temperature sensor) and make sure they're working properly. Using the chart above, measure the resistance between the two electrical pins. You can either backprobe the connection or pull the connector straight off and probe the sensor itself. The ECT is directly below the throttle body, where the air enters the engine. The IAT is at the top of the intake plenum, on the driver's side, forward of the vacuum line tree. Both sensors look alike on the outside, both are round and have 2 wires going to them. On my Explorer, both sensors died and told the computer that the engine was always warm, which caused troublesome cold starts.