It does AOE damage off pykes EXP and AD. It only gets CDR per level up of the ability. You get the same Damage increase level 11 putting a point into R or into any other ability, for example. (your probably going to want the 20 second CDR on R at 11 thou.)
If the enemy has hp = or lower then the Damage, they are hit with a Execute mechanic that ignores shields mr and armor.
When an enemy champion Dies on the X after it spawns, Regardless of if it was Pyke who has done fatal damage to them (or indeed even done damage as there is a delay between cast and then damage delt to give folks a opportunity to flash or react), The Gold generation and Reset kick in, the ability itself does not need to be what kills a champion. Pyke himself will generate 
inside his own inventory and the champion who killed enemy receives standard kill gold. This interaction can be buggy, some times you can get your cut but it will not reset the ability.
Edit. items consumed by Pyke do not count for the Gold shared Tracker in his R tooltip.
When ever any part of the X deals damage to an enemy champion, Pyke blinks to the center of the X. This can be used for mobility in a pinch, its specifically useful when surrounded by multiple champs in multiple directions. you can dash to the far side of an enemy with R, then use EW for mobility to survive. Its some time needed to just survive, not casting it and dying is worse then casting it and living, you will want the $ when ever you can but its better to be up and helping your team then dead and spawning with R. If its a miss, or if its cast on minions or monsters, Damage is done to all valid targets but pyke will remain at the location of casting.
the X changes directions in relation to pyke, and it is an X shaped hit box.
Hope that helps ya understand it better.
