As an officer, is it possible to have a Primary and Alternate MOS?

Yes, an officer can have more than one AOC, you don't "lose" one when you get another.
It is common in the Army Guard/Army Reserves for an officer to be any basic branch plus 90A, especially AGRs.

You can but the only difference is your career path. Most positions feed into another unless you are in a 01A, generalist officer. If your want to go into a position as an MI Major then it may require experience as an MI CPT. Now to command an HHC it really wouldn't matter. It depends on what you really want to do. I would ask myself what I would want to be in 10 years. Then I would find someone who is doing that and review the career path that got them there.

This policy may have changed, but if you are a direct commission into your basic branch and did not complete a military academy, or ROTC, you may be stuck in that direct commission for the duration of your career, however long that may be. As an example, I was directly commissioned into the SP corps. At that time, I was informed that I would not have the ability to change branches even within the AMEDD community. As initially stated this may have changed or I was given bad info what seems like 100 years ago. Good luck!