Have been away over the long Easter Weekend -
I think John has answered the alternating movement / chit draw questions - but it is perhaps the most different aspect of the game for people to wrap their heads around. We debated keeping it or going back to the standard "you move / I move" turn sequence - but in the end, as people got used to it - the chit system adds so much color and uncertainty to the game we all think it is the best way to go.
So, to answer your question again to make sure it is clear: If you are a "1" initiative leader, you can activate and move only on the "1" chit or on the "0" chit draw. They may come earlier in the turn or later - but they are the only chits that enable you to activate - so you have fewer options as to when you move.
If you are a "4" initiative leader, you can activate on a 0,1,2,3, or 4 chit - so you have the option to move on any chit when it is drawn.
The counter-offensive is to prevent armies from being "pinned" by only 1-2 units attacking them - preventing them from moving. One can declare a counter-offenisve, blow away a small pinning force and still be able to move and attack. It happens rarely in the games we have played - players know if they want to pin an army, they have to commit a big enough force to actually pin it. . .