Yes, normally it is safe to assume the DIMM sockets are fine, as a breach in the factory solder are almost unheard of, but failure in the RAM modules themselves would be MUCH more common by comparison.
You can test all of the RAM at the same time, it might save you some time as the only time you want to test one at a time is if you get errors when testing them all at the same time, otherwise you don't have to go through the entire ordeal four times. Remember to not touch the (as best you can manage) the copper connection on the bottom of the RAM module (the part that fits into the socket) as you can easily discharge static from you body into these, and RAM is one of the most static sensitive components on a PC.
When switching cases, it is very helpful to make yourself a clean workspace and carefully remove everything one step at a time laying the components down next to you in the order you remover them with the screws for each component separated (I fold paper into a little "dish" and place each set of screws in the paper next to the components.
Obviously, removing the motherboard itself should be done with the most care, as the screws are tiny, and if you have the protective anti-static sleeve the motherboard came in, place it on that to avoid static discharge, If not, then try to place it on a surface such as wood that is not a conductor of static electricity.
When you place the components back into the new case, just reverse the order in which you removed them and everything should go in nicely. Remember if you use a screwdriver with a magnetic head be careful what you touch with the head, and always try to keep one hand on the chassis of the case to keep yourself electrically grounded.
Take your time and everything will go fine, it's not as complicated as most would make it seem (or as I make it sound
Good luck.