Designing a PDA w/o $\epsilon$-moves and $\leq 2$ states to accept an $\epsilon$-free CFL by final state
Indeed, as you say, a PDA can accept by final state or by empty stack. The two acceptance criteria can be effectively translated into one another. Main ingredient of that translation is to clearly mark the bottom symbol of the stack, so the PDA knows when the stack will be empty. In that way it will not accidently block or accept when not planned.

True, given a CF grammar you can construct a single state PDA, accepting by empty stack. In that simulation the contents of the stack correspond to the part of the string not rewritten in a leftmost derivation. If the CFG you start with is in Greibach Normal form, then the PDA can be constructed to be "real-time" i.e., it has no $\epsilon$-moves.

The point is to combine both ideas. Thus mark the last nonterminal. When it is removed (not replaced by another nonterminal), then move to the second state, the final accepting state (at the same time emptying the stack).

For the CFG from your example $S\to aSB, S\to aB, B\to b$ this can be obtained by changing the grammar a little and making a "marked" copy $\bar X$ of each nonterminal $X$ to indicate bottom of stack: $\bar S\to aS\bar B, \bar S\to a\bar B, \bar B\to b, S\to aSB, S\to aB, B\to b$. Production $\bar B\to b$ is the final one. $\bar S$ is the new start symbol.