When the mage successfully casts the control action spells when do they have their new meat puppet act? Do they get an immediate action through the puppet? Do they issue the command and on their action and the puppet acts on their own initiative or on the mages? If no command is issued does the puppet just stand still or do they act on their own and if so what happens when they act before the mage issues a command?
With
Control Actions, you operate the puppet on your initiative, and it takes a Complex Action to do so (SR5, p.292-293). The puppet does nothing if you're not actively controlling it.
With
Control Thoughts, the puppet acts on its own initiative, following the last instructions it received (SR5, p.293).
For these sustained spells does the spell immediately end when LOS is broken or is the mage just prevented from issuing commands while the target is out of line of sight?
In general, you only need line of sight to cast a spell; you can sustain it indefinitely even after the subject moves out of view. As a practical matter, you probably need line of sight on the subject of a mental manipulation to give instructions. And if their Logic + Willpower exceeds the Force of your spell (or you're playing under Missions rules), your control will only last a few seconds anyway.