When registering for college courses, can I use both TA and my Chapter 1606 benefits?

If you plan on going to school for more than 4 years, I would recommend looking at other options than burning through your GI Bill as a "Top Up." You can use student loans, and then request SLRP as a reenlistment incentive. Many states also have a Veterans Grant that covers 100% of tuition at public universities. There are probably other options as well. Once you figure out a long term education plan, you can plan which incentives to use when.

You can, but I would also look at other scholarship options as well.

You can use your GI Bill, BUT, you have to pay out of pocket first then the VA reimburses you. I was in the same boat, I burn through my TA very quickly because my job allows me the time to do them, so I went to the Education Center to do all of that.

Go to the Education Center, they will help you submit a request to the VA and the VA will send you a letter allowing you to use your GI Bill while on active duty.

Then, for every class, you have to pay out of pocket first and then submit a document (forgot what they needed) for reimbursement from the VA.