Is an employer in the state of Texas required to reimburse gas or pay mileage to an employee that he has instructed to drive to five locations in the city to deliver letters during working hours? Which by the way is not the normal job duties of the employee.

No, there is no such requirement in Texas or any other state except California.

What is the mileage reimbursement rate in South Carolina? I work with a government agency and I am being paid 31.5 cents for the use of my personal vehicle. Shouldn't it be concurrent with the national average with is like 42 cents? :confused:

Again, unless you are in California there is no requirement that you be paid anything at all for mileage. IF your employer chooses to provide such reimbursement, it is entirely up to them how much to provide, up to but not exceeding the Federal rate. If they want to provide less than the Federal rate, that is perfectly legal. They have no legal obligation to pay anything at all, let alone being limited to the full amount permitted under the law.

If my employer chooses to reimburse at a rate below the national IRS rate, can I claim the additional expense as an unreimbursed work expense?

Yes, if you meet the requirements.

BTW, the employer CAN choose to reimburse in excess of the IRS-approved rate, but any excess amount would be taxable and subject to all employment taxes.