I don't know of any special method to find chassis/suspension noises other than to (a) bounce the car with its wheels on the ground, or (b) put the car on stands and bounce the suspension, one corner at a time. In either case, one must (a) listen for clunks and/or (b) look for relative motion between pieces that should be fixed with respect to each other. Sometimes even a very small relative motion can result in a large noise.
For your RH front shock, the easiest way to tell what's gone wrong is to take a look. If you have the required metal sleeve it should be visible from directly beneath the bottom end of the shock. If not, or if you can't tell, then removing the shock with its bracket is fairly easy.
As a safety matter, any worn or damaged hardware should be replaced with Grade 5 or better parts -- do not use cheap Grade 2 stuff as it will fail.
You can replace whatever is worn without needing to buy new shocks, provided that if you change rubber parts on one side, you also replace the equivalent parts on the other side. For a short-term repair, any rubber bush that fits should be adequate.
A front bar was standard on the GT, but neither front nor rear were standard on the JH. The lower suspension arms are identical except for the position of the bracket that accepts the vertical link from a sway bar. For the JH, if these brackets faced the front, one would have to swap the arms from side to side so the brackets would face the rear of the car.
There are at least two vendors for the sway bars generally seen on JHs, and AFAIK their brackets are not interchangeable. If you determine which brand was installed, you could contact the manufacturer for replacements. More likely, though, you'd have to fabricate new brackets, or measure your sway bar's diameter and find aftermarket brackets and bushes that will fit (check Moss, TRF, J.C.Whitney, the various racing vendors, etc.).
The original 'Metalastic' bushings were a medium-hard black rubber, while the replacements now being provided by Delta are a considerably harder black urethane plastic. Sometimes you can tell the difference between the two by poking the bushing with an awl or scribe. Visually, the replacement bushes have flanges, but so did the original bushings for the front upper arm, and the differential.
Most likely your bushings are either the ones installed when the car was built, or were replaced by original type bushings long ago. You can mix-and-match bushing brands without compromising safety.
Balancing out the relatively low cost of new bushings and the fairly extensive amount of labor involved in changing them out, I would advise to change all the bushings on the car's front, or rear, at the same time -- though it is possible to change things one pivot point at a time (but always on both sides of the car!) if absolutely necessary.
Hope this answers your questions.