Are air gaps required? If so can I get away with mounting one high under the sink? If not, will hooking the drainage hose high under the sink suffice?

The air gap is there to break a siphon from occuring. Without an air gap, the water draining down the drain can drag the rinse water out of the dishwasher.

Does everyone have an airgap? No. Do you have to have one? You should have one. Can you do one under the sink? Don't to cut a hole in the sink or counter, huh? The problem with an undersink air gap is that if it ends up venting water, as they are often known to do, it will just spill out into the sink cabinet. Do people do it anyway? Yes, will a spill bucket to catch any water.

A sink mounted air gap is highly recommended. Using an alternative method, while not ideal, is sometimes used; although, it may be in violation of local building codes, and may result in water damage, poor dishwasher performance and/or the voiding of your DW warranty.