1. The rules state that shaken units that are also out of command will be completely unable to move, since both incur the penalty of reducing a units movement points by half.
There are actually four ways to have a units movement halved:
1. Unit is shaken
2. Unit is out of command
3. Unit moves and fires
4. APC loads or unloads
Does this mean that if any of the two above apply, the units can't move?
ex. A shaken APC that loads cannot move
ex, APC that loads out of command cannot move
Is it then true that units cannot move and fire under any of these circumstances: unit is shaken, or unit out of command or APC loads/unloads?
You are correct on all accounts. Simply put, whenever a unit that already has a 1/2 Move condition on itself, cannot voluntarily undertake an action that would incur another 1/2 Movement penalty.
A Shaken APC cannot move after loading
An OOC APC cannot move after loading
A Shaken unit cannot fire on the move
An OOC unit cannot fire on the move
An APC cannot fire on the move after loading/unloading
2. Do shaken CO units rally other units normally, applying the LCR and RRR modifiers normally? Do they rally themselves applying the LCR modifier?
Yes, a Shaken CO is able to rally itself and other formation units normally, without penalties of any kind. And yes, the CO unit may use the LCR bonus on itself when rallying.
Does the +1 hill movement penalty apply to moving down hills?
No, the movement penalty only applies to units going uphill.