does any subset of symmetric group generated by repeated multiplication of 3 or more permutations can be generated by only 2 of them? (trivial cases of identity and single transpositions aside)
is there a way for finding such set of two permutations from bigger sets? (naive brute-force search aside)
(and, well, what is correct name and terminology for this problem?)
If I understood your question (1) correctly the answer is certainly no. For example, in $\;S_6\;$ , we have that
$$\;H:=\langle\,(12),\,(34),\,(56)\,\rangle\cong\Bbb Z_2\times\Bbb Z_2\times\Bbb Z_2$$
and this group cannot be generated by less than three elements. You can easily generalize the above to any finite number of generators.