Is there a limit on the number of stakes we can lay?
There is no limit, however the defenders only have so many knights. If you can work out how many there are, you know what your maximum needed number is.
I'd like to know how easy/possible it is for the enemy to clear Fascine.
Once complete, it is not possible for the defenders to clear the fascine.
How do Mantlets and Siege Towers work? For example for a force of 1 troll, 1 orc and 4 goblins with 1 siege tower and 4 mantlets, how much archery fire do we need to worry about during an attack?
All archers on the walls get one attack against the attacking units as they advance to the walls. At this point, any units using mantlets will be invulnerable to archery fire at this stage.
When your forces reach the walls, they will start to climb. Any archers not engaged in melee will get one attack per round, and it takes a full round to scale the walls. If the scaling units are using a siege tower, they are invulnerable to archery fire at this stage. If there is not enough ladders/towers for all units to climb at once, the spillover will wait their turn at the bottom of the walls. They may also be targeted by archery fire, unless using mantlets.
Thus in your example, there are a few different ways it can go down.
If the troll pushes the Siege Tower alone, whilst the orcs and three units of goblins use mantlets, then the trolls and one unit of goblins will be vulnerable during the approach. Once they arrive, Two units will be ascending in the siege tower, whislt the remaining four are protected by mantlets.
Alternatively, you could have the Orcs and Trolls each using a mantlet (requiring one strength each) and working together to move the Siege Tower (requiring three strength total) whilst two units of goblins advance behind mantlets and two units are exposed.
Finally, when awaiting their turn to climb, Goblins may suppress defending archers using their own range weapons, at a ratio of two goblins per one archer. Suppressed archers can not perform ranged attacks. Archers in towers cannot be suppressed.