I think that Deepwood Motte wouldn't be too hard for Mance to control and protect from the Redwyne fleet, mainly because the redwyne's would have to sail pass the iron born and at this point in time I just can't see them defeating them. Remember that the only reason Victarion was defeat in the Greyjoy Rebellions, was because Stannis out thunk him and I can't see Paxter Redwyne pulling that off, especially if Euron is there. It is most likely that the Ironborn would never loss control of the Western Coast of the North if Mance was to come down and it would mean that Mance would most likely have a significantly diminished domain, but I don't see the Ironborn having the military might or the sheer force of will to try and fight Mance and his 100,000 man army.
Onto White Harbour, I still believe that Mance could effectively hold the city from the Redwyne fleet. If I remember my facts, the extent of the Tyrell forces was something like 80,000 troops with the Redwynes in possession of about 200-250 War galleys and possibly 2-3 that many merchant vessels, which would be for transport only in the event of a full scale assault on the North. Now when Stannis attacked Kings Landing he had roughly the same amount of War Galleys, yet that wasn't enough to transport his army of 20,000, most of whom went on horse back to kings landing rather than ride in the ships. Considering Mance would most likely be holding, though with much difficulty, Moat Cailin, this would mean the bulk of the Tyrell-Lannister force would be trapped in the south. Transporting troops by sea with the Merchant Vessels to the west coast would be considerably difficult for the Redwynes, because they'd have to contend with the Ironborn and probably suffer considerable losses. So it's most likely that the Redwynes would only be able to transport as many as 30,000-40,000 if they used all their vessels and sailed at White Harbour. Considering how well defended the Harbour of White Harbour is, with multiple catapults and scorpions adorning the walls, I can see Mance and a force of say 60,000 warriors being able to hold it, or at least weaken the Tyrell forces to such an extent that even if they took White Harbour, they wouldn't have much luck holding it for very long. Also considering Manderly had at least 23 massive War Galleys, I believe that they could be used, ineffectively of course, but to enough an extent that they could make life considerably difficult for the Redwyne fleet and cause some losses, or at least clog up the harbour and make it difficult to transport troops in.
I will relent in that if they really wanted the North, I think the Lannister-Tyrell army could take the North from Mance, though I believe the causalities on their armies would be far to great for them to legitimately consider doing it. Plus there is also the threat of the Ironborn who I don't believe would give up whatever possessions they have in the North, which would make it equally as costly for the southern forces to attack the North via the West. However supposing that the Ironborn no longer have a presence in the North, I agree that Mance would most likely loss. Though I still think he'd do some much damage to the southern forces that they'd never risk attacking him, especially if he's commanding 100,000 experienced wilding warriors. Whether they're cover in heavy plate or leather, I think their natural hardiness and tenacity would probably serve them well against any threat from the south.