You should start the process 3-6 months prior to moving. You never want to wait until the last minute to see what you qualify for, or if you need to make some lifestyle adjustments to get the home you want.
Your qualification will be based on your monthly income and your monthly obligations.
It varies based on the program you qualify for.
That answer is no, you have the power to choose.
There are several factors to consider, including interest rate, mortgage insurance amount, terms of your payment, and down payment requirement.