NO NO NO. Do not connect a capacitor. Do not run the AC as long because it is pulling the voltage down so badly. Big voltage drops mean wires are carrying current beyond their capacity, and they are getting HOT. Something is very wrong and it could burn your house down and kill you.
Starting loads for motor driven items like compressors are essentially a dead short until the motor gains some RPM. That is why the lights dim when it starts. This is relatively normal, and wires and breakers are designed to deal with short-term (10 second) overloads like this.
The fact that your lights never get back to full brightness means that the voltage drop is not due to the starting load, but instead is due to the continuous load of running the compressor being too much for the wiring. The lights being dim means that the voltage is below 110 volts. This is because the current load is so heavy that the resistance of the wires in your walls is consuming power and generating heat.
This is bad. Normally, running this way for more than a few minutes will trip the breaker, because as voltage drops, the current required to drive the load increases. The breaker senses the increased current draw and eventually trips, protecting the wire from heating up too much and starting a fire.
If the breaker isn't tripping, and your lights are visibly dimmer than they should be, then you are in a dangerous situation. It is very possible that the breaker is too big for the wiring. The reason this is dangerous is because the wires will get hotter and hotter when they are carrying too much load. The breaker won't care how hot the wires get and will NOT trip until the current goes over the breaker's capacity. If the breaker is too big, your house can burn down and the breaker will never trip. Even if the cord to the AC is not hot, that doesn't mean that the wiring in the walls is not hot.
Someone may have put a 20 amp breaker on a circuit that was wired with 14 gauge wire, which should only have a 15 amp breaker on it.
Other things to note:
1. Do not run the heavy load off an extension cord of any kind. Extension cords are generally not intended for high duty-cycle use at their stated capacities. There are exceptions, and they are specially marked for heavy loads.
2. It's possible that the wiring and breaker are fine, but there is a high-resistance connection (something not tightened properly) somewhere. The most likely place is the outlet itself. If you can switch off the breaker, you can check the connections on the outlet to make sure the screws are tight. If the plug is wired with the wires just shoved into little holes on the back of the outlet, then it's not going to be good at carrying heavy loads.
3. I would HIGHLY recommend you don't mess with any of it. Find someone who knows what they are doing. Death is VERY permanent. You are correct about your amount of knowledge - it is dangerous.
Please forgive if this seems pushy, but the situation you have is not safe, and you should NOT dig into it with your level of knowledge.
Tim