a) It was invented by Grothendieck, see EGA I, Springer edition, especially chapter 0, discussion of representable functors.

b) Quite possibly is a shortcut for $Hom$. Sometimes the letter $y$ is used (for Yoneda). The trouble is when you are considering the representable functor defined over several categories, e.g. a category and a subcategory.

Bonus: If you, instead of considering contravariant functors $\mathrm{Sch}^{o} \to \mathrm{Set}$, use covariant functors $\mathrm{Aff} \to \mathrm{Set}$ the notation used in EGA is $h_X^{o}$. Perhaps the reason is that Yoneda's map is contravariant in this case.

Let me answer the questions in order.

a) It was invented by Grothendieck, see EGA I, Springer edition, especially chapter 0, discussion of representable functors.

b) Quite possibly is a shortcut for $Hom$. Sometimes the letter $y$ is used (for Yoneda). The trouble is when you are considering the representable functor defined over several categories, e.g. a category and a subcategory.

Further evidence on a) The notation is already on SGA 3 and 4. There are several exposés by Grothendieck in Henri Cartan's seminar from 1960/61 in which he explains his point of view of Teichmüller's space through representable functors in the analytical category and he uses the notation $h_X$.

I an not aware of anyone else using these ideas at that time. Cartan's seminar is available at numdam:

http://www.numdam.org/article/SHC_1960-1961__13_1_A7_0.pdf

See also Bourbaki seminar, exposé 195 (February 1960)

http://www.numdam.org/article/SB_1958-1960__5__369_0.pdf

Bonus: If you, instead of considering contravariant functors $\mathrm{Sch}^{o} \to \mathrm{Set}$, use covariant functors $\mathrm{Aff} \to \mathrm{Set}$ the notation used in EGA is $h_X^{o}$. Perhaps the reason is that Yoneda's map is contravariant in this case.