I paid $725 in tuition but this is not listed in box one on my 1098-T? Is this a mistake?

You would have to ask the school that question.

But, the 1098-T is only any informational document. The numbers on it are not required to be entered onto your tax return. 

If you claim the tuition credit, you do need to report that you got one or that you qualify for an exception (the TurboTax interview will handle this)

You claim the tuition credit, or report scholarship income, based on your own financial records, not the 1098-T. In the 1098-T screen, click on the link "What if this is not what I paid the school" underneath box 1. You will then be able to enter the actual amounts paid. 

Yes, because if you paid tuition, it should have been reported in Box 1 of the 1098T. In this Turbo Tax link it explains how you can bypass this. In the 1098-T screen, click on the link "What if this is not what I paid the school" underneath box 2. You will then be able to enter the actual tuition amounts paid.  Make sure you keep receipts and other documentation proving that you made this tuition payment in the event you are audited by the IRS.