If I walk far enough my altitude is also sure to change, so you might consider walking a three-dimensional activity, but this is a position subject to debate. Two directions are orthogonal if there is no way that motion along one of these directions could result in motion in any of the other directions. Walk east or west all you like, you'll never see any north-south change in your position. That's why walking is not normally considered a three-dimensional activity. That one option results in upward motion while the other results in downward motion seems more a function of the surface of the earth than in any choice on my part.
Orthogonal directions are always perpendicular to one another.
The thing with dimension is that it refers to more things than just the number of independent directions. Any two measurable things that are independent of one another can be considered dimensions. Identifying the appropriate perpendicular directions in a kinematics problem is the first step to solving it. Naming the other directions is open to debate, however. High school physics courses tend to concentrate on two-dimensional problems.