NY Resident doing out of state consulting
1- Would NY be able to claim Nexus just due to me being a corporate officer in a NV corp?
2- Would I be able to deduct mileage from my home in NY to drive to NJ, being that there is no corporate headquarters?
3- How would payroll work? Does NV corp need to withhold NY or NV taxes?
4- Does anything change if I would use WY instead of NV?
From the description of your business I see several problems with having your business organized anywhere but New York (or maybe New Jersey if this gig is pretty much permanent or you plan to have more of those in that state).
Basically, people form companies in Nevada, Wyoming, Delaware etc for very specific reasons. Often times an online business would choose to be incorporated in a state like Wyoming or Delaware, and if you plan to build a cutting edge new start-up and you plan to raise a few millions you might want to consider forming a Delaware corporation for that purpose.
Without going too deep into why those states are good for those particular cases and not for others, I would say that often times a Delaware corporation would register to do business as a foreign entity in California, New York, or any other state where it has actual physical operations. Having your state of incorporation in one of the incorporation friendly states would be helpful in intra-company issues such as dealings between shareholders etc, however for the purpose of running your business in your own state you would have to register there anyway, either as domestic or foreign entity. So you might as well register as domestic and save time and money.
In your case since you plan to take deductions specifically related to your geographical location, such as car expenses, maybe office, your own salary if you plan to take one (you might as well just take member distributions, but at that point I would suggest you to consult your accountant for further professional advise), I think you might want to form your company in New York to begin with.
Finally, just FYI - forming and maintaining your company in Nevada is quite costly nowadays, and there is a steady flow of companies originally formed in Nevada being domesticated in Wyoming (that state offers very convenient domestication procedure). This is practically why we omit Nevada as one of the incorporation friendly states and mention Wyoming and Delaware only.