1
Map and region are interchangeable, most references to map should say region. A region is contained on one sheet of paper.
2
all creatures above size 1 are large creatures. When mounted you the rider are threatened by smaller creatures for purposes of your attacks and you can be staggered, pushed or knocked prone (although you do have height advantage). The mount however isn't threatened and can move with impunity through threat areas. the rider counts as being all the spaces of its mount.
This is a little barebones for space and because mounted combat isn't that common, but I think a really cool rule would be to let riders only be threatened on size 2 creatures by enemies that threaten both their spaces. Might be a little finicky but it makes mounted characters hard to threaten rather than easier which I like.