Can a character do an additional (not bonus) action in their turn if they didn't move? Like attacking two times without moving? I remember something like that from the handbook, but I can't find it anymore.
On your turn, you can move a distance up to your speed and take one action. (PHB p.189, "Your Turn," emphasis in the original.
This section is followed by one which discusses bonus actions, which you clearly understand are limited to one per round, no matter how many sources/features might grant you a bonus action.
Nowhere is there stated an ability to forego all movement in exchange for a second action.