I've been learning to play NP in bot games and I still don't know when the best time to use his ulti. Do I cast it on weak creeps close together so it then bounces to other waves after the one near the area where I casted it?
You can use it:
a.) During team fights
b.) To push lanes
A - You want to cast it in some lane away from the fight because the ult does more damage the more creeps it hits before reaching/passing through the target area. Say there's a fight breaking out at your bot T1 tower. You cast your ult on an enemy creep wave top. When the ult finally bounces to the enemies bottom, it's doing more damage than it would have if you cast it where the fight was breaking out.
You don't always have the luxury of doing it this way and can sometimes miss out on a kill if someone slips into the fog of war before the bounces get to the area you wanted the bounces to hit at, so keep that in mind. Sometimes you NEED to cast it at the fight. You'll get a better feel for that as you play NP more.
B - Ideally, if pushing lanes, you want to pop the ult with a maximum number of creeps on screen. That simply means that ideally, you see 3 lanes of minions that your ult will bounce through. If you don't see at least 2 waves of minions on the minimap, you wait until you do to use it. Remember, the more bounces, the more damage being done and the more impact the ult has. Also, the better the chances you get some creep kills and income coming in from the ult.
Does shallow grave counter LC's duel and does Shadow Demon's Purge remove the bonus damage stacks?
Not sure about Purge, but yes Shallow Grave counters/supports duel, depending on what team he's on.