You need to hire an attorney for a summary administration. In addition to getting the deed transferred to your name, the attorney will ensure that the homestead benefits are transferred to you. If you don't do this, you will always have chain of title issues with the home.
You should contact a local probate attorney for guidance and direction in your situation. You will want to be sure that the title is properly transferred and that any remaining issues are properly resolved.
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Since you're not a surviving spouse and you didn't own the property with your father as a joint tenant with rights of survivorship (all assumptions based on the facts provided) you must contact an attorney to have the will probated. Plus, you need the probate process to preserve homestead rights to avoid losing the home to pay off creditors.
Yes-you can achieve this through summary probate.
The mortgage can be transferred to you.
St. Pete Attorney
The answer given does not imply that an attorney-client relationship has been established and your best course of action is to have legal representation in this matter.
You are able to handle this through Summary Administration, which can be accomplished very quickly in Florida. You should have no problem taking care of the mortgate at that time.