What is the grace period for federal student loans?
For federal student loans the grace period is after you graduate, leave school, or drop below half-time, you would have a six month grace period and sometimes 9 months for Perkins Loans.
Is it possible to make a payment to another loan servicer if I don't want to use Great Lakes or Nelnet?
Yes you can. You can also get a 0.25% interest rate deduction when you  participate in automatic debit.