I think the drills do have a speed, affected most often by how much power is available. It's my understanding that the vanilla upgrades affect the yield, like production modules. That should stack with the yield of whatever resource is being drilled, because the module/upgrade yield is applied by creating a secondary production bar for the drill.
Also, does placing a drill over more ore patches speed up drilling?
No. The drills pick a random resource entity every mining cycle, and the amount of ore that comes out can be "fuzzy". Mining at 200% yield will always yield 2 pieces. At 250%, there is a 50% chance to yield 3 pieces.
Can drills "poach" from each other if their active fields overlap?
Yes.