Does changing the lens of a camera allow me to take higher resolution video?

I'm looking to buy a 12.3 megapixel Nikon D5000 dslr. The problem is it only shoots 720p video. I was wondering if I buy another lens if it might allow me to shoot higher resolution video.

No. The resolution is determined by the processor chip inside the camera. Changing the lens won't change the resolution.

No

Video only uses a portion of the 12 mp camera sensor.

In the case of HD video, 1280x720p uses less than 1 mp (you can do the math) Even Full HD (1920x1080p) only uses 2.2 mp

Here is what you need to know about shooting video using essentially a still camera with a video feature

http://www.luminous-landscape.com/reviews/camcorders/cam-cam.shtml

No but don't worry about it. 720 is going to be fine. Most of what you watch on TV is compressed down to 720 anyway.