As far as tyres are concerned, your friend is right. You'll find that your ride on-road is easier and quicker if you put on slick or semi-slick tyres.

With regard to the rim, are you sure you need a new one? Is it that the rim itself is damaged, or is the wheel just out of true? If the latter, it may just be that the spokes need tensioning correctly. If you know how to do that for yourself, fine. If you don't, and a new rim is really required, you're going to need some expert assistance anyway to build up the new wheel.

Can't help with cheap bikes, I'm afraid, I don't know the US market. But at that money I think you'll be looking at buying used if you want something reasonably light.