Do parallel, angle, triangle, area etc still apply in Mobius band?

Away from the edges, the geometry of a Moebius band is locally Euclidean, so the usual concepts and theorems of geometry all apply. The issues involving edges are qualitatively no different than if you were trying to do geometry on a disk instead of a plane: non-parallel lines may still never meet because they fall off the edge.

On the other hand, if you could form a one-sided surface with no boundaries, such as a Klein bottle, then things would be different. For one thing, the sum of the angles in a triangle is no longer 180 degrees. Since there is no obvious way to have a Klein bottle without some places being more tightly curved than others, this gets messy really fast, but the concepts are not a lot different than those of spherical geometry or hyperbolic geometry.