Can my employer enforce my non compete?

It all depends upon what the agreement says. There are legal conditions the agreement must meet in order to be enforcible. If your employer just threw something together, it's likely that it wouldn't be enforcible. If they had a lawyer write it up, in such a way that it is enforcible, then, you guessed it, it would be enforcible.

The link below highlights some details about Texas non-compete agreements that you might want to check before signing the agreement.

edit: If your dismissed former co-workers are taking clients with them, your boss would have had you sign a no-solicit agreement as well. If that wasn't part of the package you signed, then it's likely your boss just threw together an amateur agreement, the kind that likely wouldn't be enforcible. Have your lawyer review it to make sure though.

Quite possibly, yes. Have a lawyer read the agreement.

I have a part time job in the same industry doing the same thing I do at the company.
In EVERY jurisdiction, even those that 100% refuse to enforce a non-compete agreement, that is grounds to INSTANTLY terminate your employment.

A non-compete agreement deals with working as a competitor AFTER you leave the company, and without reading the actual terms, NO ONE can tell you if it is enforceable in ANY given jurisdiction.
An agreement not to open a competing business across the street and take clients with you for 6 months is enforceable EVERYWHERE.
An agreement never to work in the same industry anywhere on the planet for the rest of your life is un enforceable everywhere.

DOn't sign it, for now.
Go to HR and let them know about the other job, and see if they want to negotiate giving you full time or simply letting you go. Don't ever sign anything that works against your interests, and don't ever sign something in the belief it cannot be enforced. Go speak to them, and take your lumps. If they don't want an employee that works in two places, you're going to have to find another place. You don't want to be dealing with a non-compete anyway, even if you think you'd win. You'd still be in court, as a matter of public record, and that is never good for your employability.

Yes.

Yes, but wouldn't be able to claim damages suffered up until the date you sign the non-compete. In other words, you have no liability until you sign and continue working for the competitor. If you quit the other job, on or before the same day you sign the non-compete, you are golden.

Sure. You are probably the reason everyone has to sign an agreement. That will at least give him cause that is better than just firing you this Friday. Why is it that you never discussed this with them when you were hired?

Yes, he can enforce it. You should have told him of your other job. He might have made that an exception. Since you didn't, you're scr*wed. Employers almost always sue over a non-compete when a past employee breaches their contract.

I don't think such agreement would be legal unless your company is under DOD contract.
A non disclosure agreement would be legal.