boeing247 asked: Do most commercial pilots only fly planes by a certain company?

That depends on the airline, and what planes they have. If you work for Southwest it would be Boeing 737.
TWA had several choices of planes to fly. Boeing, Douglass, Convair or Lockheed. Pilots bid on what they wish to fly in seniority order. The most senior pilot gets the first choice and the junior pilot gets what's left over. The way my career worked out I flew Boeing planes. I almost went to school on the Convair 880, but stayed on the 727 since it was more pay and better working conditions. Pay is usually based on gross weight, something left over from the airmail days.

Once you go to school on a certain plane you are usually frozen there for 18 months, unless the company moves you. This is due to the cost of training and until the 757/767 came along each plane was so different that you were normally kept to one plane at a time. Some pilots were dual qualified, but that was a lot of work to keep both planes up to date for books and landings. At one time as an instructor I flew the 727, 707 and the 747...that was a ton of work.

Lou