Nah, you can sight either one in fine at 25m. I have the TSR-100 and have never missed having adjustable elevation at the rear sight. The TSR-100 has a dual aperture rear sight with apertures at different heights, which takes you from approximately a 25m zero to a 50m zero with just the flip of the aperture (which is all you need most of the time for a .22). You have more elevation adjustability with a TSR-200, but with the 100 you still have plenty of elevation adjustment at the front sight. The 200 requires a tool like a FMJ bullet tip or a ballpoint pen or something to adjust it anyway, just like the front sight.
Now on an SKS or AK or something that doesn't have nice MOA-incremented clicks at the front sight, I recommend the 200. But for one of TS' .22 kits that comes with the AR front sight, I don't really see much advantage in having elevation adjustment at the rear as well. The fact that the 100 is cheaper seals the deal for me.
Hopefully you've figured out that to mount a sling on your Ruger, you will need to drill your stock and install sling swivel studs as well as sling swivels. For Appleseed, you will want to get some 1 1/4" swivels so you can use a GI web sling, which will let you use all the sling techniques taught at Appleseed (which is an important part of the instruction). They are sold at the Appleseed store, or on E-bay. You want the web sling that is standard issue for the M-1 and M-14. You should be able to pick one up for $15 or less.
When you go to the Appleseed, bring both the 10/22 and the Mosin. You can practice the fundamentals and do most of your shooting with the .22, and then bring out the Mosin later in the weekend to apply your skills to it. Don't listen to those guys who complain about the recoil... you can handle it just fine if you listen to what the instructors are saying about turkey neck, cheek weld, and pocketing the stock. A .22 is great for learning the fundamentals on a budget, but there are some things about rifle marksmanship that cannot be simulated by a rimfire, such as recoil, and the necessity of building a solid position that can be maintained through recoil, so having both will be good for your development as a Rifleman.
And yes, Rifleman scores have been shot with Mosins, including by yours truly. I don't know what kind you have, but the full length barreled 91/30s and Finn M-39s are the best for this type of shooting, since they have long barrels, which gives them good sight radius even though they have tangent sights. The carbines are more difficult to shoot accurately. But use that factory sling as a hasty sling, get smooth with those stripper clips, and execute the fundamentals, and you'll be fine. Appleseed is not an equipment race. It is very much a run-what-you-brung type of event. It is also not a .22 shooting club. I know center fire ammo is expensive, but I appreciate the guys who show up with center fires to make it sound like a real firing line!
btw I'm an Appleseed shoot boss and the state coordinator for Oklahoma.
Hey, good to see you on here, 4bfox! Hopefully I'll see you soon on the trail.