Replacing a threadless fork?
 

Should any 1 1/8 threadless fork work with any bike that has that kind of fork?

AND, will all the little plastic ring parts that are on the top & bottom of the stearing tube be fine to use on the new fork without purchasing anything else?

Well, no.
All threadless forks aren't interchangeable.
1 1/8" is a quite reliable standard. But first the fork needs to have the right steerer tube length - or longer - to be possible to fit it to the frame.
Then the fork legs needs to be right too. The axle-to-crown distance needs to be about right, and the rake - forward extension - needs to be about right too.
Then the fork needs to be compatible with the brake type you intend to run.
Dropouts have to match the axle type in the hub you want to use.
Width by fork crown needs to allow the type of tires you want to use.

But yeah, all forks with 1 1/8" steerers should be able to use the same headset.

Except the ones with different crown race diameters......

By little plastic ringers are your refering to the spacers? those should all fit.

Which would that be?
I've only seen the ISO/JIS mismatch on 1" (threaded) steerers. And referring to a tapered as 1/18" only would be leaving things out.