There is a Theorem of Tits in Comptes Rendus of Academy of Paris  "Systemes generateurs grupes de congruence" ...(early 80's) where he proves this (the finiteness of index of the group generated by powers of the root group generators) for any Chevalley group of $Q$ rank at least two. For the symplectic group, the root group generators coincide with the generators that you have listed.