1. Simulation suddenly changes speed rapidly in 11th frame, and speeds up too much… How to make that change gone? I'd like to be just a bit faster/more explosive in the beginning, but slower towards the end.
2. How can I control the timing/lenght of sim? To specify beginning phase, middle one, and final when pieces are getting out of camera?
first of all, your bomb RBD's collision volume is set to implicit sphere display Collision Guide Geometry to see how it looks. As you can see this will create only one bounding sphere your whole particle system you are using as a collider. This means that the sphere will grow enormously fast and will push everything out
You can use magnet force with animated active parameter instead of object pushing geometry from inside
to retime the cached simulation use the timeshift SOP after the dop import in model object, delete default expression ($F) and use your own curve to timewarp the simulation, be sure to uncheck Integer Frames checkbox