Your steps are pretty hard to follow, but if I understood them correctly, you got it right.
Normally, Aimed Shot (AS for short from now on) has a cooldown of 3 turns. Which means you can use it every 3rd turn, with 2 turns with no AS in-between.
Shooting Spree triggers and reduces the cooldown of AS on every sniper rifle kill (unless there's no cooldown) including the AS kill. So if you kill someone with AS, the cooldown becomes 2 instead of 3 turns already. So AS every 2nd turn, with one turn with no AS in-between, as long as you hit and kill with them every time.
Yes, it is possible to use AS every turn if you manage to kill another enemy with regular sniper rifle attacks or Snipe in addition to the AS kill. It happens from time to time, but not nearly as often as you would expect.
Let's go through an usual scenario with a Rapid Spearhead:
Turn 1: Snipe + AS. Snipe kill doesn't reduce the cooldown in this case, since AS is not on cooldown yet and you have to start with Snipe. AS kill reduces the cooldown to 2 turns.
Turn 2: AS coldown down to 1. Now you have 2 shots to take. If the first of those shots kills an enemy, the other one can be an AS this turn. If not, you have to wait for the next turn. Keep in mind, it's not often that you manage to kill a full health enemy with one regular sniper rifle shot, unless it happens to crit or the enemies are really weak.
Then there's Limited Temporal Increment, which reduces all cooldowns by 1 turn every 3 turns. Or every 2 turns, with Future Orientation. That makes it significantly easier to use AS every turn. But that's a whole different matter, and it's not that easy either.
