Delaying and freezing cloth sims -
As far as I'm aware the the cloth's physics can't be delayed or 'paused' directly but there is a workaround.
This example has the cloth lying on an invisible plane to begin with, thereby holding it's fall until the required frame.
It's released by switching the plane instantly down to a location directly below the bed and over a single frame range.
NOTE - The cloth's setting may have to be changed (keyframed) to suit when it's lying on the plane and when on the bed.
The cloth will fall and settle onto the character and bed but will most likely want to keep wriggling. To avoid this it's visually switched to a 'frozen' copy of itself.
To freeze -
Save the Blend file at that frame firstly and then 'Apply the cloth's physics modifier. This will give us a frozen duplicate which MUST be cleared of all physics and animation!
Save the project again but as a temporary file.
Reload the previous save and append the temporary. Copy paste the frozen mesh into the original scene.
You should now have a soft cloth plus an identically shaped hard-mesh version at that particular frame.
At that frame, visually switch over to the hard mesh version in the Outliner. (Keyframe the camera and eye icons so we see the duplicate, not the soft original)
Play and check that the cloth falls, settles and remains stationary.
Save that as the project Blend file.
When the character sits up, at the frame where that starts, visually switch back to the soft cloth again.
At the frame where the cloth settles again, do as above. Save the project file firstly and at that frame, 'Apply' the mesh and write that out to another temporary file. Append that back into the 2nd project file, etc etc.
This can all be simplified considerably when done in the VSE, however this would be a bigger challenge when the character is to move during the cloth's frozen frames as we have in this example.
The dummy character used here is copyright so including the Blendfile isn't possible, however it's all in the procedure. A matter of freezing, substitution, and switching their visibility.
Should you feel it necessary to have the Blend file, I can redo it with a cube character.