Moving the location of a bathroom sink
Yes use one for both the waste and vent. If the existing sink doesn't have issues draining, your vent will be fine. For the waste, just "T" the two sinks together and join them to the existing drain/waste pipe. Do the same for the water supply for the sink. Remove the existing cut-off valves and put a tee on the stubbed out pipe with the opposite end of the tee pointing at you and the leg of the tee pointed to the new sink. Run a new piece of pipe from the tee over to your new sink and put a 90 on the end. Connect your shut-offs for the old sink at the tee and for the new sink at the 90. Of course you'll have to run a hot and cold line. You don't have to, but I would replace the existing cut-offs with new ones while they are off. Good luck!

Normally you would stub out a drain for each sink and then have a p-trap directly under each. This is what is shown in the photo , before the p-traps are installed. It is permissible to have one trap, but if you do, it has to go below the tee. 
I'm old school and don't use Shark fittings. I'm not sure if they are acceptable in a sealed wall or not. Hopefully someone else can help you there.