The rules say to use the terrain type at the centre of a hex. A couple questions:
i) One hex actually has a river at the centre but most of the hex is woods. I would suspect woods would be the type used for combat
As stated, the hex is woods for combat. Just curious, since river hexes always run along hexsides where did you find a river in the middle of a hex. Are you sure this isn't a coastal hex? Rivers that enter a body of water often widen out and then become coastal hexes.
ii) How could you ever have a battle on mountain types since there are no mountain hexes on the original maps (perhaps expansions have them) and only flyers and mountain folk can even inhabit them? If you fought in a mountain pass what would happen?
Most mountain terrain follows hexsides and therefore limits movement. Mountain passes are only along hexsides so they don't affect combat, only movement.
I wonder if a variant of permitting combat modifier based on entry hexside terrain could work? (I might be out ot lunch on that. Probably disbalance the game).
What would be the purpose of this? Hexsides limit general movement and movement into a battle hex and so serve a purpose in combat already. Hex terrain affects combat. Seems simple enough. And what about hexes with multiple different terrain types on hexsides? In your varient, which do you choose? It just wouldn't make much sense without a bunch of new rules.