This is an answer to the first question.
Let's indicate with $r(K)$ the ribbon number of a knot, i.e.the minimum number of ribbon singularities needed to realize a ribbon disc spanning $K$. We have $$ r(K)\geq g(K) $$ This is shown by Fox here:http://ir.library.osaka-u.ac.jp/metadb/up/LIBOJMK01/ojm10_01_08.pdf
Mizuma has shown that under certain conditions on the Alexander and Jones polynomial you can assume that $r(K)\geq 3$. This is Theorem 1.5 here:http://ir.library.osaka-u.ac.jp/metadb/up/LIBOJMK01/1782ojm.pdf It is a very special situation and I don't think that much more is known in the general case.
Maybe it is worth noting that given a band diagram for a ribbon disc one can add a fake ribbon singularity near each singularity and then eliminate both with a tubing operation. This produces a Seifert surface whose genus equals the number of the original ribbon singularities in the band diagram. The definition of a band diagram and a picture of this trick can be found here:http://etd.adm.unipi.it/theses/available/etd-07062011-061816/unrestricted/Polynomial_invariants_of_ribbon_links_and_symmetric_unions.pdf (pag. 27)