Are soldiers required to have a Government Travel Card?

yes

Unsolicited testimonial ~
I've been banking with USAA forever! Whenever I used my credit card out of the norm, it would be declined. Locally, Nationally or out of Country. I always bought something stupid right off the bat and call them and explain that I'm on TDY orders, their reply was "What would you like the new credit limit to be?" If I said $2,000, $5,000 or $20,000 they would punch it in their computer and I was covered..

Yes. I was forced to get one in 2013. After being screwed over by using it to pay for commercial air tickets (that I was directed by CTO to do) and ended up having to pay $1100 out of pocket for the honor of being assigned in Korea, I let my card expire. I still have the card, it just isn't valid.

Yes, it's actually public law, not just army regulation. There are a number of exemptions to the requirement, but the only likely one you would fall under is infrequent traveller, 2 times or less in a year. Even then, your commander can still make it a requirement for readiness reasons. My advice to you is to go onto the TRAX website and take the training there seriously. Then talk with your Agency Program Coordinator if possible or someone at your unit that is knowledgeable about the use of the GTC. It is a very effective tool when used properly.

In my Unit, yes, we are all required to have a GTC.
There are still a few personnel who do not have one and have gotten away from it for an extensive amount of time. The issue with that is the fact that Supply cannot track down every person in our Unit to get a GTC. If I had a choice, I would not have gotten one.