The first time I ever crewed in a race was on a Chrysler 22 or 24, over 25 years ago, and we won the regatta, so I think it can be a fun racer, but, at this age it probably needs the old bottom paint stripped to make it competitive.
The availability of parts depends on what parts you need. Some parts are standardized, and can be purchased or ordered from sailnet or other marine suppliers. Parts that are specifically made for the Chrysler 22 will probably have to be fabricated. I doubt that you'll find a bone yard that has a stock of Chrysler 22s. With a little ingenuity, most parts that can't be replaced can be fabricated by the home handyman.
I would definitely not hang her from the ceiling. Without looking at them, I'd bet the two stern eye hooks and the bow hook are not designed to hold that much weight, and, if the garage or other structure only has attic space over it, it also might not be strong enough. If a ceiling isn't designed to be weight-bearing, it probably won't support the weight. If you do suspend it with straps, however, they should be positioned where there is a bulkhead inside to reinforce the boat. If properly positioned, the straps shouldn't hurt the boat.
Good luck with the project!