Hi Flo,
Welcome to the forum! You've got a very nice American Eagle Luger in original finish.
A large retail dealer like Simpsons can give you an idea of the insurance value of your pistol. Check
these listings...
Luger collectors value a Luger as close to original factory state as possible. Original finish, all matching parts. You have these things in your Luger, and it is very collectible.
Remember as you approach stabilizing the red oxidation taking place on your pistol that there is always going to be some level of red oxide present on an older gun. This is actually evidence of the original state of the finish (which was rust blued, with strawing done on several small parts - the golden ones). It is easy to go overboard on "cleaning up" your Luger.
I would be very conservative about doing anything with the wooden grips or magazine bottom(s). They can be gently cleaned, but part of the aged appearance of the wood will be affected by cleaning. Oil based soaps designed for cleaning wood can be used, but they can also be too harsh if heavily used.
G. T. on this forum (Gerald Tomak) is an expert on magazine repair and restoration. I personally would not dis-assemble a magazine, but would be more comfortable sending it to him for any needed work.
So, best advice is to go slow, gentle and just do enough to stabilize the finish and prevent future red oxidation. Certainly some can be slowly removed to help with the stabilization process, but there should always be some red deep in the blue-black finish.
Also remember that many of the new relatively exotic chemicals sold for gun cleaning didn't exist when our Lugers were originally made and maintained.
Marc