For example, lets say the SP is besieging Minas Tirith but has not yet taken control of it. Can the FPP declare the fellowship in Minas Tirith to heal the corruption or make it safe in the stronghold?

FSP isn't stopped by besieged Strongholds like Companions are and can heal in a Stronghold under siege, so I see no reason why they couldn't declare in a sieged MT. But SP would get bonuses to Hunt rolls if the FSP tried to move while MT was still sieged (or worse, taken) and SP could also play "For the Dark Lord" combat card for sieges. FP could of course counter with their "It's a Gift". At least, that is if IIRC the card names correctly.

What Dam meant in "bonuses", is that the SP would get to reroll failed hunt dice: At least one (since the area has to have a shadow army for the stronghold to be sieged), possibly two (if there are Nazgul with the army). If the stronghold falls, and the fellowship moves again without being declared in a different area first, the SP would have an additional reroll.