If you haven't been in contact with a DA yet, then you could be weeks out still. Or not. There really isn't a way to know for sure. As for home delivery versus picking up, since the car is already in Dedham you basically need them to truck it to you or you'll pay MA tax. Otherwise you could pick it up in NY. Since you DA hasn't contacted you yet, though, the vehicle might not yet be at a distribution / delivery center. The date and location you see on the Tesla site is simply a placeholder, and you shouldn't believe anything it says.
The only reliable way to know when the car is coming is that the trucker will call you on the day of delivery to confirm you are or will be home. There are stories of trucks just showing up and the driver calls from nearby, though, so once your DA gives you a general idea when the delivery might be, stay close by for a day or two.
As was stated, the truck driver will have a form and an app for you to sign noting any damage. Their form doesn't have things like paint defects though, so any swirls, nibs, dust caught in the clear coat, panel alignment issues, etc. they likely won't have on their sheet. You need to be ready to take a TON of photos when your car gets dropped off, and take copious notes. My experience from beginning of October was that my paint needed some correction work which I had done by a professional detailer near me in Hooksett. Panel gaps were effectively a non-issue, and my biggest complaint was sticker residue that I cleaned off my windshield with a flat razor blade.
My interaction with my DA, on the other hand, was a completely miserable experience and I'm glad I won't have to deal with that for the foreseeable future. Stay on top of your DA, put together a list of documents your bank needs (if you're financing) and make sure you have them. Be sure to pay before the truck is supposed to be scheduled, because the trucking company can not bring the car to you until you've paid in full. Have insurance on the vehicle the day before it's due. Make sure your DA over night's the registration paperwork package so that you have it before or when the car shows up. In NH you can not legally drive the vehicle until it's registered, and you must have the paperwork from Tesla to register it. My car sat for 5 days while I waited for paperwork and then town hall to open on a holiday weekend.
Finally, you can reject delivery, but you may want to carefully consider that option. If you reject, you may not get a replacement before end of the year, so you'll miss out on $3750 of the potential tax credit. You'll also go back into the queue, and likely mess up someone else's delivery date in the process. It might be much easier, if the problems are simple, to take detailed photos of the issues, mark them on any paperwork you can, and to call your DA right then and there to see what works best for you. If you take delivery of a car that needs paint touched up, that might be a much better option than waiting a month or two for a new car. You can probably get Tesla to cover fixing it somewhat quickly or paying for a certified/reputable shop near you to fix it. After curing for a couple days, car paint should be hard enough to be out in the snow and salt.
