For your first question I believe this paper may help a bunch. It has a 6 bit combinator calculus that is also an UTM. Also it has a universal combinator that seems to have size 7 with one element given what you want. They call it Zot. http://arxiv.org/pdf/cs/0508056v1.pdf
I am not sure if you can say or prove that there is a minimal combinator. The paper would suggest it would have to be at least be less than 6 bits.