Let's keep these brief...
Siege Rules
A walled fortification of any kind can be held as part of a siege. Non-walled sites, even towns or markets, cannot be held unless they have some sort of fortification to besiege - defenders must either flee them or fight until there are none left or the foe is defeated.
A siege happens when an attacking force rather than attacking camps outside the area controlled by the fort and prevents transit in and out of the fort. An attacking force of any size can attempt this. If a fort has unrestricted access to a port (either because the port is not held by enemy troops or because the fort is situated in a port) then both land and sea must be blockaded for a siege to take effect.
Once a siege is underway, the defenders will slowly starve and deplete whatever rations are left in the fort. Unless the fort has recently been besieged, forts start with a number of provisions equal to the fort size (i.e. 1 unit for Walls, 2 for Forts, 3 for Castles etc.). Every regiment within a fort will consume one shipment of provisions per season. If the defenders should end a season without sufficient provisions, they must either surrender or immediately attack the besieging forces (and try to flee).
Whilst besieged, defenders may upgrade regiments present in a fort but may not recruit new regiments there, although regiments can be recruited outside the fort if messengers can get through. Getting a messenger out requires a successful Scouting check (that is, a Scout unit can be used to improve its chances and Scout units can be used to oppose it) that succeeds on a roll of 3 or better. Unless allies exist outside who will pay for recruitment, messengers must also carry the Ducats used for recruitment with them - so capturing a messenger will put those Ducats in the hands of the captors.
Merchant adventurers can try to smuggle provisions into the fort from other provinces - any edible staple will do, and the market will pay full military+staple demand bonuses for it. The defending government may also pay a bonus at their discretion. Smuggling goods past a siege requires a successful smuggling check (as smuggling contraband), following the usual smuggling rules. It is possible for merchants to try to bribe NPC officers in the besieging army to help as they would bribe customs officials. Other than breaking the siege, this is the only way to bring fresh provisions into a besieged fort.
As things are, Castle Sleen will run out of provisions at the end of Spring, provided the port is successfully blockaded.