Yes, as long as your Priority Date is current (according to the Visa Bulletin). The more important is the timing of filing your I-130 and your...

You do not need your own Birth Certificate. You, as a Petitioner, need proof of your U.S. Citizenship, along with other documents required.

Your Employer's recruitment activities should be based upon the Prevailing Wage Determination.

Your prospective employer (U.S. employer) may seek to become a sponsor for your Green Card through the required processing.

I do not see any apparent issues for your Employer seeking to file an L-1A Petition for you and for you with an entry to the United States with...

Once you commence your employment with your EAD by pending AOS, you will no longer have your F-1 Status.

Once you file your H-4 Change of Status, you have an option to leave the school or continue studying, if it is your preference.

Unfortunately, nobody can predict how the Visa Bulletin will move. The Visa Bulletin sometimes moves backwards, too.

Please be honest to disclose all your history and information in your Visa Application. The final discretion is up to the Interviewing Officer.

Generally, a person in L-1B status is authorized to stay up to 5 years (upon approval).