Sleeping 2 in a Geo Metro?
Those are little cars. I'm having a hard time imagining it. I might suggest spending a night in it to see if it really works for a good night's sleep. Just laying in it for a few minutes to check it out isn't gonna give you the true picture.
You've gotten lots of good advice so far that you should take.
It's not impossible that this car is OK. But I have a hard time believing that a car at this price will have good tires. If your traveling partner has good reason to trust this mechanic based on recommendations or good prior experiences, then I see no reason to take the car to NY just to have it checked out. But if the mechanic's judgment is questionable, I would sure take it to another mechanic to be sure.
Personally, I think any car should be given a thorough look-over by a good mechanic before a long road trip. At the very least, a thorough checking of all fluids and hoses, an oil/filter change, a brake check, an alignment check, and a tire check are always on my list even though my current car has never given me any mechanical problems. So that's the least I would do with your car. I'm sure it could use a tune-up, at the very least.
Private campgrounds always have showers. Sometimes, especially if the park has very low rates to begin with, they require quarters for the shower. Some national/state/county parks have showers, but not all, and some don't have potable (drinkable) water so you'll want to make sure you have a good supply of your own water for cooking, tooth-brushing, drinking.
You seem to be very inexperienced when it comes to road trips. It's too bad that a short trip to check out the car and to gain experience doesn't seem to be something you have time for. I wouldn't let that deter you but I think you might want to do a lot of reading on this forum to get some ideas about how to do a road trip safely while having a great time.
Here are some things you should consider:
* I think a AAA membership (or some other road service) would be a good investment. I don't travel without it for those just-in-case situations.
* Access to emergency funds in case you do have a break-down or some other problems you hadn't anticipated (or just because the trip ends up costing you far more than you had considered)
* Travel with a cellphone and charger. A CB is a good investment as well since you will probably drive through some places with spotty or non-existent cellphone service.
* Have a person that you check in with daily and let them know what your plans for the next day are.
* If you are doing some hiking, it doesn't hurt to leave a note in your car stating where you're hiking, when you plan to return, and listing emergency contact information. I would leave this face-down on the dash so it's easily seen but can't be read by just anybody. You really only want emergency personnel in a worst-case scenario to see this information.
* When hiking, be sure you have The 10 Essentials (easily find with a google).
* Make sure you have at least basic emergency stuff in your car like basic tools, working flashlights with extra batteries, flares or emergency triangle, blankets, water, a basic first-aid kit, etc.
* Oh, and make sure your spare tire is in good condition!
Since money seems to be a major concern, I might suggest that you read this to help you plan your budget. Scroll down to the 7th post.
Hope this all helps!