First question: Story telling! Also, the main issue usually isn't just applying damage to a cube. The issue is knowing where to apply it. Also, beaming isn't that easy due to the usual "shields block beaming" issue. There are also explosives and other chemicals shown as being too instable to be beamed, so this might be another reason. As seen in First Contact, the Borg just wiped Starfleet's blockade with obviously minimal or not too serious damages. Once the Enterprise arrives, all they essentially do is coordinating attacks on specific points of the cube (they obviously tried this before, but probably never hit the right spot). Although, to be honest, I'm not sure why this strategically important information wasn't shared/passed along earlier (or it's been ignored? Maybe just more story telling involved...).
Different approach to this question: Why don't they just use beam technology for attacks, like beaming away essential parts of the warp core compartment or anything like that? I.e. you don't have to destroy important things by exploding them, just remove them by beaming?
Second Question: Their shields block beaming just as usual and known throughout the series. However, Borg usually won't waste power on shields unless entering a fight or being engaged. Most often beaming on board a cube happened before fighting, after fighting or while the cube has been damaged. As far as I remember, the only difference is that Borg are able to enter other ships with drones while the shields are still (partially?) intact.