I only ever go Grasp against someone who I know will only take short trades and who I know I can't consistently kill during the laning phase e.g. Renekton or Camille. The point of Grasp is to help keep up during those short trades, and (sometimes) the enemy will avoid taking trades with you since they know Grasp hurts if you're able to auto attack with your passive.

In summary there's not that much of an advantage going Grasp instead of your regular Conqueror but in lanes with lots of short trades its more useful. Taking Conqueror even in the aforementioned matchups is not a bad idea since its universally Aatrox's best choice in most games.

Depends on the matchup. I think that build is alright, it has a lot more damage compared to the basic Eclipse -> Sundered Sky build. The main gimmick is that you have much less survivability (with no Eclipse shield) and you rely way too much on your ability to hit Q sweet spots. I'd say that its really good against squishy team comps, especially if you build full lethality after core items.

Personally I'd opt to build Eclipse in case you're unsure what to go for as its pretty reliable and quite consistent. But like I said it depends on the enemy comp. If they're mostly squishy then I don't see anything wrong with going Shojin + Cyclosword.