Can you install a new faucet with an extra spray outlet and use the dishwasher vent hole in the sink thereby connecting the dishwasher drain directly to the disposal?

Not sure what you mean by dishwasher vent hole, but you probably mean the air gap fitting. Some dishwashers connect directly to the disposal, with the tube going up a little to prevent stuff from going back to the dishwasher. But as for removing the air gap fitting, if you have it already, it may depend on where you are. California plumbing code requires a dishwasher air gap fitting above the flood level rim, even if you have a disposer, but I'm not sure if the national code also requires. In any case, if a code requires it, it may be a good idea not to remove the air vent. I would check your local code. You may want to consider a faucet with a built in sprayer instead.