Do you think it is a good idea to include your company name in each page title after the main title? For example: New York Cars - Car Company of America New York Sport scars - Car Company of America NY Car Parts - Car Company of America
It helps make your title unique and brands your name with the idea. It won't really help you rank a lot higher, but it won't hurt. Although, if you have a really long page title and then append your company name to the title, SE's may truncate the title.
Depends, is you company name also the main keyword or keyword phrase you are trying to rank for? If yes, it adds some value to each of your web pages containing it in the topic for that specific keyword\key phrase.