Are these a normal lace shoe or the 'bungee' type?

No its not normal lace up, the lace is on the straps on the side of the shoes

Will these drain or just hold water when doing off-road obstacle course races

Hold water..... look at the addidas climacool for draining. They have the air pockets and will drain but usually mud races all shoes hold mud and some water

Thinking about buying these to wear in a kitchen. Thoughts?

While they are very comfortable, on oily surfaces they are slippery as ever. Comparable to ice skating. I will still buy them because of the comfort. I am just careful by slippery spots at work.