Should I enter half my mortgage interest under deductions and half under rental prop expenses if I rent out half my house?

Assuming that half of your house was rented for the entire year, yes, you would claim half of the mortgage interest on Schedule E and half on Schedule A.

Just pay close attention to where you enter the information.  You should come to Schedule E first.  Pay close attention to the entries to see if there is a message about letting TurboTax take the personal portion to Schedule A automatically.  Then when you get to Schedule A, read those screens carefully to see if the amount has already been entered.  Otherwise, it is easy to have the deduction doubled by accident.