As a reservist, can you use TA and GI Bill Selected Reserve at the same time or alternate between the two using TA one semester?

Yes, you can use both at the same time, assuming that you meet the conditions for both.

No, they stopped this a few years back. If you had CH31, Post 911, or the REAP then you could but not Chapter 1606. I would recommend saving the GI bill until you deploy and become eligible for more benefits. However, If you cannot do that use your TA for ass much it will cover then use the GI for the other semesters. I believe as long as you are not paying for the same course with both programs you are in the clear but speak to a VA counselor at your school for specific guidance.

Ask education center at local base. Although you mention being a reservist, the counselors there should know if you can or cannot "double dip"