I don't think so; being static is a very restrictive condition on manifolds. The paper doesn't seem to talk about this, as far as I can tell.
No. Parallel transport does imply "moving" vectors from one event to another; that's what it's for, to allow you to "compare" vectors at different events by moving one of them from one event to the other.
OK I'll take another try. A particle can travel on a closed geodesic but it's world line will of course be unbounded.
Then wrt closed geodesic:
a) it's world line would be a set of point worldlines together comprising a tube.
b) it is a single entitiy whose world line is a tube.
c) It is an abstraction that can't really be said to have a worldline.
so does parallel transport include a velocity term?