I just realized I never answered the last part of your question.
1. How would you press in the plastic dome?
I use my single stage press. Of course you go easy, but I get them in with-out incident. My neighbor loads his with a Lee Hand Press.
They loaded like any other projectile.
2. How would you keep the shot in the dome to load them?
The capsule comes with a plastic base that is snaps into place once the shot is loaded.
I bought some and I think I was five plastic bases short. I called Speer and they sent me a whole new box of shot capsules.
Inside Every Bright Idea Is The 50% Probability Of A Disaster Waiting To Happen.