Oddly enough, one of the things I wrestle with the most in Blades is how to adjudicate Controlled positions. Specifically, if a PC pushes themselves for an extra dice and decide to withdraw and try a different approach (if they roll a 1-5), do they get to keep the extra dice on the next roll, or do they have to start from scratch?
Also, if they roll a 4/5 and withdraw to try a different approach STILL from a Controlled position, and then roll another 4/5, can they again try yet another approach, still at a Controlled position?
In Band of Blades, it is explicitly written that you keep any bonus dice from DBs and assists when you press on from Controlled to Risky after a 1-3 and re-roll. But not the bonus dice from Pushing.
In BitD, afaik there is nothing like this rule. But you can choose to allow the player to keep the bonus if it fits the fiction of what he’s doing.
For the second question I would say yes, why not, nothing prevents it. Except the new approach chosen (even for the first re-roll) is not necessarily in Controlled position.