Hello Elkslayer. Welcome to the wonderful world of the Handi-rifle!
You are on the right track so far with your cleaning. These fine little guns generally need a good cleaning prior to use. Seems you hit the correct areas. Some even recommend removing the butt stock and spraying the reciever thoroughly with an appropriate gun cleaner.
Most folks have had problems with military ammo sticking in the chamber. I experianced this with my .223. Due to the success of others on this site, I now use the Winchester white box 45 gr. found at Wally World, Gander Mtn, Bass Pro, etc. Some folks have fixed the sticking problem by using J-B Bore polish. You'll finds several threads about that on this site.
The O-ring acts to float the barrel. Many have used this with success. The general recommendation seems to be to fire your rifle first to check if you need to. I've learned to allow the rifle to properly cool between shots to achieve the most accurate results. Additionally, it seems that resting the Handi rifle on the receiver rather than on the forend also has a positive affect on the accuracy. Resting on the forend seems to cause a binding action at the receiver
If your rifle came with the factory mount it should have a hammer spur. The spur will allow you to use low-med mounts depending on the size of your scope. If no hammer spur, you may have to use high rings so the scope will clear the hammer. Reviewing the previous threads, you will find that a variety of mounts are used. There are pro's and con's. Read whatever you can and make the decision thats best for you. I have had good luck using Leupold Rifleman high mounts for my handi-'s without the spur. I have also used the Weaver Quad mounts with good success.
Again, Welcome to the site. The moderators and contributors are a great bunch. You will find that there is a wealth of knowledge available here.
EVOC ONE