If I go to another company will my emplyer be required to pay me for those earned hours?

Yes, it's legal but if the amount of time spent is more than insignificant, you can demand compensation for that time. If he doesn't pay, you can complain to Wage & Hour.

I'm not sure what your question is. In general, you may be protected under the Family Medical Leave Act or Americans with Disabilities Act depending on the cause of your absence, but I would need more information. I suggest you speak with an attorney. You are required to be paid for any hours you work, but I am not sure what you are asking in the first part of your post.

It sounds like you have two separate questions in your post. The first I would ask for you to rephrase, as it is unclear as to what you mean by "all those hours". As to the second question, forcing someone to work while on sick or vacation leave sounds highly unusual and is probably not permitted. However, as is with most legal matters, it is the details of your situation that will determine whether or not those hours are compensable and under what law. An experiencd employment attorney would be able to provide you with many of the answers you are seeking.