Have you downloaded the Ames manual yet? The noiseless typewriters are in volume 3. The download link can be found on several places on the net, I am pretty sure the typewriter database site has a link available somewhere.
   I took the carriage (and the subunit called the shift assembly) off of my Underwood noiseless standard a couple of years ago. But Remington and Underwood were different companies, even though Underwood licensed the noiseless design from Remington. They (Underwood) often still used their own design for parts that weren't specific to the noiseless action, so my experience may not help you. I also have a broken Remington model 10 noiseless standard in the back room that I haven't started work on (escapement problem).

I just typed a lengthy and thought-provoking reply to a user's problem in the maintenance board. And, when I clicked the "preview" button, the page blanked and sat there with a little spinning icon up in the menu bar. All it said in the lower left corner was "transferring data from www.google.com". I went out into the kitchen to do something and it was still locked up when I came back so I just killed that tab in Firefox and lost all of my beautiful and heart-rending words. I am sure I'll never be able to write such a perfect ode ever again!
     It probably has nothing to do with this message board software. Likely that the internet gods are just toying with me today. But I wanted to rant somewhere about it.

If you type the same letter over and over, for example, the uppercase "N", does it keep a straight baseline, or does the baseline go up and down?

To add to Uwe's post: I would suggest using a lightweight card stock as a backing sheet. I have found that various retailers in my local area, mostly car dealers, send me candidate backing sheet flyers almost monthly printed with their most recent sales promotions. Check those out since they come free in your mailbox. I have found that you want a card stock that is lightweight enough to roll around the platen without holding the paper bail out from the surface of the platen. This provides the greatest cushioning without the hassle of messing with multiple sheets of backing paper. Just your typing paper, and the card stock backing is easy to handle.

Curious. I don't have a Royal 10, but I do have the KHM and the KMM models and both of those have a worm that turns and holds the worm gear on the spring drum shaft. And I can't find it on the pictures you posted. I can see the worm gear on the shaft, that's the black sleeve with the slanted teeth and the set screw on the end of the shaft that leads out of the center of the spring drum. But there is no worm to mesh with it that I can see.
   I'm probably just not familiar with the way the Royal 10 model is built since I don't have one. So if it's hiding down there somewhere where I can't see it then feel free to disregard this post.