If a string is unorientable, why is the a string with reversed orientation different from the initial string? Why do we have Kalb-Ramond 2-forms?
the statement in the question is not true. If strings are unorientable, then - by definition - the state with a reversed-orientation string is the same one as the original string, up to a sign (and/or transformation acting on Chan-Paton factors).
It also follows that unorientable strings can't carry any charge that could take infinitely many values. Consequently, the Kalb-Ramond 2-form is always removed from the spectrum by the unorientability projection.
Among the 5 ten-dimensional superstring theories, only type I string theory has unorientable strings. In modern language, type I is type IIB with a spacetime-filling orientifold O9-plane, which makes the strings unorientable, and 16 spacetime-filling D9-branes and their images which allow open (and equally unorientable) strings in the spectrum and give them the $SO(32)$ gauge group carried by the end points.
The remaining four 10-dimensional superstring theories - type IIA, IIB, heterotic-O, heterotic-E - contain purely orientable closed strings which are also charged under the NS-NS 2-form B-field. Type II string theories may be supplemented with orientifold planes - type I is an example; type IIA needs even-dimensional orientifold planes and type IIB odd-dimensional ones. Heterotic strings can't be made unorientable (or open) because the left-moving excitations on them are inequivant (bosonic string theory) to the right-moving ones (superstring).
I suspect that you have simply confused the words "orientable" and "unorientable" - rotated their meaning by 180 degrees. The "simpler" theories without projections etc. are those with the orientable strings. To make strings unorientable, one has to do extra things and identifications - or, equivalently, add an orientifold plane. It's perhaps linguistically counterintuitive that the "un-...-able" are the objects that require a special treatment while the the "...-able" ones are those that don't. ;-)