Typically, no, you couldn't perform both actions in the same turn because it represents using the same hands. Actions that require the hands are limited by the hands.
How does that differ from drawing a weapon (which is explicitly listed as an action) and attacking with it. I know it's getting down to a legaleese reading of the terms here, but I don't see a difference as long as both are called actions.
Because drawing a weapon is a specific exception and has its own section for that reason.
This is not drawing a weapon; it is grappling with a foe who does not want to be grappled but also throwing that person (who does not want to be thrown either) at the exact same time.
In this case, it is more like wanting to fire a rifle (Shooting) and hit someone with it at the same time (Fighting). Got to pick one.