Drive-off and down payment can be (almost) as much or as little as you can negotiate. I've got quotes from 6 different dealers before I leased my e-Golf on Dec. 19. The numbers I received were as little as $1,000 "out-the-door" (meaning down and drive-off combined) to as high as $4,500 --- it's a matter of how much money the dealer wants to make on the deal.
As I understand on a lease the "gap" insurance is included (but you should double check this). Meaning if the car is totaled the insurance will pay off whatever you owe to the lease company (VW Credit). But the down payment and drive-off fees, $4,000 in your case, will be "lost". Therefore my advice is to put down as little as possible, and just focus on the monthly payment between dealers.
Of the dealers I've contacted, I told the 3 whose prices were the most reasonable, that I wanted to pay $1,000 out the door, including 1st payment (this put all 3 on equal footing), then I've just picked the lowest monthly payment (in my case it's $383/month including tax, and I'm in LA).
I'd recommend you use Edmunds.com and TrueCar.com and contact the internet sales department of several dealers. Also, you can use Cars.com put in your ZIP and a radius, and it will show you how many cars are around you, what dealerships, and colors.
Hope this helps, let me know if you have any question.