My bike is using 700x42c tires. Can I swap them to thinner tires without changing rims or anything? How far can I go 38, 35?

You can use narrower tires,but they may not be what you are expecting. If you switch to a narrower tire of the same quality as the ones you currently have you will notice a difference in weight, but not in rolling resistance. One thing you will be bound to notice is that at the recommended tire pressure for the new tires, ride quality will suffer, narrower tires will have a much harsher ride than your current tires. You do not state what you hope to gain. A narrower tire on a glass smooth road will have less rolling resistance, but if the roads near you are not perfect, your current tires may well be much faster

You can, but I dont recommend it, rolling resistance can increase on all but the smoothest road surfaces with narrower tires, plus you lose out on grip and comfort

Depends on the inner width of the rim. Here's a link to a chart with recommendations.

http://slowtwitch.com/images/glinks/articles/WhatW...