To start with, you left off the dates of your H ... making is nearly impossible to see if 6 years have passed since your first day on an H.
Second, NO ... no one can start 'immediately' after filing an H-1B petition. Even with premium processing it will take 15 days.
PROFESSOR OF IMMIGRATION LAW for over 10 years -- This blog posting is offered for informational purposes only. It does not constitute an attorney-client relationship. Also, keep in mind that this is an INTERNET BLOG. You should not rely on anything you read here to make decisions which impact on your life. Meet with an attorney, via Skype, or in person, to obtain competent personal and professional guidance.
You will not be subject to the visa cap when an employer files its H1B petition.
However, you must wait for the H1B petition to be approved before you commence employment. The exception would be if you were currently in H1B status and an extension of H1B status was filed either by the same employer or another employer.