Nsxboy7,
It sounds to me like you are getting a voltage drop in the system. First thing I would do is recheck the battery terminals for tightness, and also look carefully for cranks in the terminals. Is the car starting OK, every time?
If it always starts OK, then the other possibility is your alternator is really not working correctly. Could be as simple as a loose belt. Why did you replace the battery? Has the alternator been checked out to see if it is OK?
I ask because what really seems to be happening is you are starving the car for voltage. When you put the car in park, you most likely have your foot on the brake, so the brake lights are on, drawing power. When you raise the windows, same thing. The brake lights and the window motors demand alot of power, so if your alternator is not 100%, you can dip below the proper voltage and all sorts of things may get strange.
To confirm this, you can try a few other accessories to see if you get the same result, headlights, heater blower, etc. I suspect any big electrical demand will produce the same results. If so, get a voltmeter on the battery terminals, read it with the car off, start the car, then read it again, should be about 1.5V HIGHER to properly charge the battery. Then activate the brake lights, voltage drop?? If so, check alternator belt/alternator for proper output.
Any good auto electrical shop can test it fully if you get to the point you think the alternator is the problem. They can bench test it for full output. The alternator, as the name implies outputs alternating current, but you really need direct current to supply the car with. Inside the alternator is a set of diodes, known as the "rectifier", it converts AC to DC. If a diode goes bad in this rectifier, you will not get proper output from the alternator to the car. (this is typically what will go bad in an alternator electrically, or they become mechanically bad, if a bearing and/or slip ring fails.
With all this said, please do not assume the alternator is bad, it could just be a loose belt.
HTH,
LarryB