I am a fan of leaving things be. If you start rubbing, even with soap and water and now there are rub marks, or the paint starts flaking, or whatever. It has lasted this long, I would just leave it as is and bask in your find!

Please do not clean the inside of the liner it is in fine condition as it stands. Also, when you find a sweat band, and the nape strap try to match the condition of the liner.

I also can say that it looks good. Can we see the outside of the liner?   To your questions. Perioe would be 1942/43, the correct strap will cost you around $150. A correct Airborne Chin Cup might be $400. Liner suspension add-ons would be easy to find, $50. An M1 Fixed Bale/Front Seam McCord Helmet would be appropriate, $350. An M2 $$$ Thousands!   To be honest there are enough photos on USMF that you should feel free to post any pictures you want. What Jerry and I would be hesitant to do is comment on individual specifics that the liner you have shown, which have characteristic you would want to see. 

And another question.  What is the best way to clean/ preserve it.  It came out of the woodwork and, as one can see, still has 80 years of stuff baked into it, and some rust residue on the mostly intact external paint

Just one more in case it helps dating it....

Good points, and equally, I don't want to "give away" the tells of a good one vs a fake one, as fakers watch this forum too. But from what I can see, I do like it.  

Thank you, Mr. Jerry!  What "give-away" photo should I post?  Don't want to post every angle to avoid pics being "lifted"...  

Here's a Roosevelt's SS patch made for Fury.. It's not attached to the cap, but just laying on it    

I am a longtime helmet collector, (still not an expert) but this seems like a nice early M1C original liner (more pics would be needed, as the fakers have gotten quite good and the mod is pretty simple).   As for the loose bits, I think the parts all show up, but the parts can be quite expensive when found individually. I would think a leather chin cup is what you would want on there.   I would look for an early M1C para helmet (or M2 if you have deep pockets) to go with it.