Federal tax---yes of course. You have to pay federal tax no matter what state you live or work in. FL has no state tax so there is nothing to withhold for FL. Your Florida employer may or may not withhold SC tax, but if the employer does not do that, you are responsible for paying SC tax since you live and work in SC. You might need to pay quarterly estimated tax to SC.
Yes. as champ @xmasbaby0 said, your Florida-based employer should withhold federal income tax if you are an employee. If you are an independent contractor, then — no — your company does not have to withhold tax and you will have to make quarterly estimated tax payments.
Fill out a Form W-4 and give it to your employer for federal withholding.
You can fill out a Form SC W-4 and give it to your employer to get state tax withheld.