(1) This LDO is intended to power the bandgap circuit, so there is no expectation that it will use a bandgap-derived reference at all. If so, it must not bootstrap itself, but could use the bandgap voltage to stabilize the LDO if switched in very carefully after the LDO powers up the bandgap. The bandgap has three specified voltage outputs, and one of those is preferred, if a reference is used.
(2) Yes, the LDO input voltage is the 3.3V avdd supply.
(3) Yes, I think the dropout specification was for the LDO as it was specified before deciding that it would be used only to power the bandgap. You can ignore the dropout constraint.