Rearrangements; Carbocations; E1 Mechanism
1) Will the methyl shift? I'm guessing the answer is an emphatic yes.
  Tertiary carbocations are more stable than secondary carbocations.


Yes, if that is the only way to generate a tertiary carbocation and the C-Methyl bond can more or less line up with the carbocation p orbital, then the methyl would likely shift.  If there are other alkyl groups attached to the carbon initially bearing the methyl group, they could shift too.


  2) Will there still be some secondary carbocation left over? I.e. will
  there be both major and minor products? I'm guessing yes


An equilibrium exists between the various carbocations.  At 300 K, an energy difference of 1.4 kcal/mol results in a power of ten difference in the equilibrium (e.g. from 50:50 to 90:10).  So it really depends on the energy difference between the various carbocations involved, as well as the rate they are captured or proceed to the next step.

Finally, if there is a hydrogen on the carbon adjacent to the carbocation center, then elimination to create a double bond becomes a competing pathway,