I'm asuming that you have a stock "sporterized" Saiga that takes 10 round polymer mags as opposed to a pistol gripped, hi capacity rifle converted by Hesse or someone else. The stock Saiga's have a feed ramp built into the mag that sits higher than a regular feed ramp, so adding one to the rifle won't work. It seems your ammo is getting hung up on the feed ramp. Here's my thoughts.
1. You may be "riding" the charging handle forward with your hand as you try to chamber the first round. This reduces the full spring force on the bolt and then it doesn't have enough power to push the round past the feed ramp. Since fired rounds do not hang up, but manually charged ones do, This is my first guess. Try pulling the charging handle all the way to the rear and then simply let go, allowing the spring to close the bolt.
2. The feed ramp in the mag may need to be filed down a little bit. Does it jam with different mags or only one? Hollow point ammo does not have a "point" at the tip like FMJ, but a little flat space. If this flat tip touches the feed ramp first instead of the side of the bullet it will stick. If you lower the feed ramp with a file JUST A BIT to alloow the tip to pass over the top of the ramp and contact the ramp on the side of the bullet rather than AT the tip it should feed fine.
The mags may also adjust thenselves over time with more rounds fire through them.
3. If you have a high capacity rifle, that is another story. I have four Saiga's, some stock, and some I've converted myself, so I can also help you if it's a hicap. Post back here with your findings and we'll go from there.
Stock Saiga's use a special trigger group, so a Red Star Arms one is out.
The doubling sounds like the disconnector is not catching the hammer as the bolt cycles. you may be using underpowered ammo that doesn't cycle the bolt all the way to the rear, or it may just be that you have to pull harder on the trigger to rotate the trigger group high enough for the disconnect to catch. Unload it, open it up, and play with the action and you'll see what I mean.
Hope this helps!
Edited to say "you beat me to it!"
so is it a hicap rifle?