Questions from my first scenario...

[ComradeP]

2) A HQ can still pass it's command test even if it's parent HQ is not on the map, having a complete chain of HQ's only serves to improve the chances that your HQ will be in command when it takes the command test.

To add to that answer, an off-map HQ can still be in range of on-map HQ's, as command range is counted normally from the position of the off-map HQ (which is usually on the full "campaign" map) to the position of the on-map HQ. The same principle applies to off-map artillery units: their range is the range calculated from their off-map position to the on-map target.

3) With this rule off you can still only move from ZOC to ZOC if the hex you are moving to contains a friendly unit.

To add to that answer, there's a ZOC movement multiplier value in the parameter data screen. If it's 0, you can never move enemy ZOC to enemy ZOC unless there's a friendly unit in the hex you're moving to, or the unit type of the unit you have selected allows it to move ZOC to ZOC at all times.

5. During my turn I can usually fire 2X with each unit - but during the AI turn it seems like it is firing a lot more than that? Like 4X at each unit?

Every normal non-artillery unit can usually fire 6 times during a numerical turn, 3 times during your phase and 3 times as opportunity fire during the phase of the other side. Maybe that is what you're seeing: first the AI's units fire in return to your units on your turn, and then they fire again on the AI's part of the numerical turn. You use MP's when firing, and the counter is reset once you hit end turn. There is no standard unlimited opportunity fire in the PzC series.