Can my employer suspend me without telling me and not get paid?

You didn't work, you don't get paid. Moreover, you knew you were suspended when you were told not to come to work.

You should still receive your normal pay check, but there is no quick remedy if they delay your payment a week or so. I suggest you wait and see. But, if you are speaking of a time that you did not work, due to this suspension, the employer is not obligated to pay you anything. In GA, they are legally required to pay you only if you do work. The fact that you state "at will employer" indicates you understand your employment relationship. If not, Google this: Georgia, Employment at will. Yes, in GA an employer may suspend you, with or without pay, without giving you a reason. If you are not working and earning wages, you may file an unemployment claim now, that will force them to tell you why. But, it may cost you your job, so I suggest you wait a week or so and see what they tell you. They may return you to work on probation or with a warning, so do not sink your ship too quickly. Patience is recommended.

No. You were told not to work for those days; you did not work and you are not owed $$$ for working. Your employer can suspend you without pay indefinitely for the purposes you describe.

Probably but it may depend on your employer's policies. You are only entitled to pay for work you performed.

If you have not worked due to suspension, generally speaking, in Texas, there's no law requiring the employer to pay you. There may be exceptions, like if you had a contract or are a union member, etc.