There are two ways to access the equity in your current home. First, sell the home. If you don't want to sell the home, the second option is to take out a mortgage to access your equity.
Yes, you can use the current appraised value of your house to help eliminate mortgage insurance on a Conventional mortgage. However, you will most likely need 25% equity vs the normal 20% for the mortgage servicer to approve the elimination.
Maybe. Paying off your student loans might help reduce your debt to income ratio allowing you to qualify for a higher loan amount. However, paying off installemnt debts like student loans and car loans might lower your credit score.