While I can't speak to how Cogmind specifically handles energy, I can (sourcing from DoomRL and the T-Engine documentation) speak to the general theory.
The system has a threshold value, above which actors can act. Any action subtracts its own duration from that actor's energy, and the system would be refilling that energy in a loop constantly. When it's the player's turn to act (that is, they have the highest energy score and its above the threshold, use some arbitrary but consistent tiebreaker if two actors are tied for energy), then this loop is paused until input is received, but the process is basically the same for all actors. There are no fixed turns, nor is there a point when turns "start", the game just goes around the table to find whoever has the required energy, and then they can act immediately.
If you must consider it in terms of turns, at least as far as implementation details go, the procedure is basically:
foreach actor
actor.energy += 1
foreach actor
if actor.energy > threshold
if actor.type == player
//handle input
else
//do AI stuff
looped forever as the game is running.
For a concrete example, let's use 100 as our threshold, set movement and shooting to take 10 energy units. Let's have a player and an enemy:
At start, let's give them a random starting energy between 0 and 9, say 6 for the player and 1 for the enemy. Eventually, after 45 checks around the table and subsequent additions to the energy totals of both actors, the system will have given enough energy for one to act:
Player energy: 101
Enemy energy: 96
Now the player can act, and right now anything they do reduces their energy by 10:
Player energy: 101 -> 91
Enemy energy: 96
At this point, it's back to the energy refill loop, until someone goes above 50:
Player energy: 96
Enemy energy: 101
Now the enemy acts, and this continues.
If, for instance, the player had an alternate fire that cost 17 energy, then this situation would occur:
Player energy: 101 -> 84
Enemy energy: 96
Wait for 5 time units:
Player energy: 89
Enemy energy: 91 -> 91
Wait for 10 time units:
Player energy: 99
Enemy energy: 101 -> 91
Then the player gets to go again (after giving all actors another 2 energy).
For a more interesting example, let's have 2 enemies, one of which takes 5 units to move. For starting conditions:
Player energy: 3
Slow Enemy energy: 6
Fast Enemy energy: 2
In this case, the slow enemy goes first:
Player energy: 98
Slow Enemy energy: 101 -> 91
Fast Enemy energy: 97
Then the player gets above 50 next:
Player energy: 101 -> 91
Slow Enemy energy: 94
Fast Enemy energy: 100
And then the Fast enemy gets above 50, but maintains energy advantage:
Player energy: 92
Slow Enemy energy: 95
Fast Enemy energy: 101 -> 96
Player energy: 97
Slow Enemy energy: 100
Fast Enemy energy: 101 -> 96
Player energy: 98
Slow Enemy energy: 101 -> 91
Fast Enemy energy: 97
And then the player gets to act again.
I hope that wasn't too complicated, though I realize there were a lot of examples there, and it should cover questions 1 and 3.
As for question 2, that varies from game to game. most roguelikes (Cogmind included AFAICT) have the animations attached to weapons fire be essentially a visual flair, but no additional time/energy/turns is/are used for the projectile itself, nor is there a way for the player to avoid a slow projectile by moving out of the way after it is fired. The only exception I'm aware of is ToME, where projectiles can hang in the air between player actions, depending on timing, so fast characters can avoid previously fired projectiles. AFAIK ToME projectiles are actors in their own right with their own energy counter.
