Yep, you will, or at least do a nail hole filling and touch up, if you just recently repainted them.

After you pull them and remove the old nails, fill, sand and paint them before putting them back on. Then you only have to touch them up when their back in place. It will be much easier than painting the entire boards after.

Only if you want them to look nice.

Yes, since you will have caulking and nail holes which need to be painted over.

You’ll need to patch and paint your walls too. Especially with an amateur removing base mold.