Mt. Washington is about 25 minutes south of me, so I'd sure be up for meeting anyone there. I've only been up it a couple times, because I've only lived up here a few years, but its a fun road, and if you've never been up the mountain, you are missing out!!! As for vehicles they allow, don't worry about it. They let me take my old ratty, rusted out Samurai up when I first moved up here. The restrictions are usually just for overly large vehicles, such as RVs and dually pickups.
The road really isn't that hard on the vehicle, as long as its in good condition. The second time I went up the mountain I did it in a '97 TJ with a 4 cyl. and 31s. Completely stock (except for the tires). Coming down isn't that bad as long as you can downshift. There are plenty of pulloffs and the scenery is awesome, so just stop a few times to let the brakes cool if they are in question. Overheating on the way up shouldn't be a problem unless you have a really crappy cooling system. Just make sure your radiator isn't completely plugged up, your fans(s) work, and the coolant is full.
When you plan to go up the mountain, leave enough time to look around at the top. There is a whole visitor's center up there with cool facts, pics, etc. One of the things you'll see up there is info about my wife's friend's grandfather, who was the only person to ever land a plane at the top of the mountain. Oh, and my wife's uncle handled clearing the auto road every spring for many years. Picture clearing snow drifts 20'+ deep with a dozer, with a steep drop on one side, when you don't even know where the road is under the snow. Talk about nerves of steel!
Jim