I've done some testing, using custom maps and ships, on the effectiveness of various modules. I may have miscounted a bit, but the numbers here should be mostly accurate.
Laser:
1 particle every 16 frames; 1.875 pps
1 energy every 30 frames; 1 eps
.533 energy per particle
Particle beam:
seems to target 1 particle per frame (could be faster)
1 energy every 10 frames; 3 eps
When trained on a single particle, the beam fires every 11 frames
.1 energy per particle slowed
Cannon:
1 particle every 40 frames; .75 pps
1 energy every 80 frames; .375 eps
.5 energy per particle
Missile:
1 particle every 61 frames; .492 pps
1 energy every 61 frames; .492 eps
1 energy per particle
MK7:
1 shot per 90 frames
1 energy every 5 frames; 6 eps
~12 particles per shot (hard to see, so this may be wrong)
1 particle every 7.5 frames; 4 pps
1.5 energy per particle
Fighter base:
2 fighters
1 particle per fighter per 8 frames; 7.5 pps
100 shots per fighter (give or take around 10 or so)
100 energy per fighter
7.5 energy per second
1 energy per particle
Grabber and Engine:
Neither seem to use any energy at all, and their effectiveness is not reduced by energy starvation. Not what I would have expected, given the "depleted" look of engines on energy-less ships.
Lathe:
1 energy per second
1 struc built per second
1 energy per particle destroyed by struc
~500 frames (not precisely counted) to lathe a ship pickup or amp gem
~150 frames to lathe an energy pickup or energy mine
~667 frames to lathe an emitter
~2000 frames for an emitter to fully heal
These suggest that emitters have the same "health" as ship pickups, but heal themselves at 1/4 lathe rate.
~3333 frames to lathe a progenitor
Healing rate seems to be consistent with 1/4 lathe rate for emitters, suggesting 5X emitter "health" for progenitors.
Some shipbuilding observations:
When building a long "snake", each hull section builds every 61 frames and takes one energy (armored hull sections take 4x as long and use 4x the energy). This is similar to (and probably exactly the same as) the rate when building multiple hull sections. When building only one module (tested with laser and MK7), one energy every 15 frames is used. This energy is not taken until the empty module part appears on the ship. My (conjectural) model for shipbuilding is:
1. Build bridge at a rate of 1 energy per 15 frames, for the cost of the bridge in the editor.
2. Fill bridge with its 50 energy storage buffer.
3. Build hull out from center in steps of 61 frames, building all connected, unbuilt hull sections in each step at once.
4. During step 3, if a module outline appears, start a 2eps energy drain until the cost in the editor is paid.
Omnis:
Omnis carry 100 energy and can mire 285 land squares using that energy. They mire once every 4 frames (note: the action of unmiring red mire and miring neutral land are equivalent for the following). Thus we have:
.351 energy per particle
2.63 eps
7.5 pps
With the Omni cannon upgrade, omnis gain a cannon which shoots at a variable rate and energy consumption depending on efficiency:
1 shot per 30 frames -> 1 shot per 20 frames (2 pps -> 3 pps)
1 energy per 30 frames -> 1 energy per 40 frames (2 eps -> 1.5 eps)
1 energy per particle -> .5 energy per particle
Amp gem fire rate modification:
Laser: no effect on fire rate
Particle beam: no effect on fire rate
Cannon: 40 frames -> 28 frames (42.9% improvement)
Missile: 61 frames -> 43 frames (41.9%)
MK7: 90 frames -> 63 frames (42.9%)
Fighter base: 8 frames -> 6 frames (33.3%)
It appears that amp gems, when they have an effect, multiply fire delay by .7 and round to the nearest frame. The rounding has no significant effect except for fighter bases, which are slightly less affected by amp gems.
Discharger:
100 frames to defeat a stunner
1 energy per 5 frames (6 eps)
20 energy per stunner destroyed
Hull damage:
Hull damage is difficult to test because a single particle may damage multiple hull sections, and I think this does affect energy usage (specifically, when testing particle damage against a square hull, the ship could stop only 86 particles before running out of energy, presumably due to multiple hull section hits.) For damage of 1 square at a time, for both particles and struc, a ship, whether made of armor or normal hull, could stop about 100 particles before the ship ran out of energy (actually, the figures were 97 for particles and 101 for struc, but I'm going to ignore this discrepancy and use round numbers.) Once depleted, an armored section took 8 hits and a normal hull section took 2 hits. This, in combination with the ship rebuilding data, indicates the following stats for a constantly rebuilding hull section (either type):
1 energy every 61 frames (.492 eps)
2 particles per 61 frames (.984 pps)
.5 energy per particle
Shields:
The effect of shields is described
here by Virgil. A shield uses 1 energy every 2 frames (15 eps).
Tactical implications:
First of all, we see that MK7s, particle beams, and fighter bases are the biggest energy hogs in the game, using significantly more energy per second than the relatively frugal cannon and missile. The cannon shoots faster and takes less energy per second than the missile in addition to its greater versatility, leaving the only significant advantage of the missile being its homing capabilities. Lasers never miss, are among the most energy-efficient weapons in the game, and are very powerful for their size, meaning that they can be quite effectively used offensively - as we have already seen with some custom ships. Also, the MK7 is energy-inefficient for use on particles; it takes 3 times as much energy per particle as the cannon, while suffering an even greater chance to miss (with its longer range and all-at-once firing). It may make a lot of sense to turn MK7's to "doppel and ship" mode when energy is a concern. The omni's mire capability is extremely powerful (as powerful as a fighter base) and is the single most energy-efficient weapon in the game (the downside being, of course, the lack of flexibility of application of that power - only particles that actually hit land may be absorbed). The omni cannon, while weak (1/4 the power) compared to the omni mirer, is essentially a double-fire-rate cannon module and thus can be decisive in particle fights.
Edit: added fighter base, corrected missile stats (thanks, Xalos!)
Edit 2: added engine, grabber, and lathe
Edit 3: added shipbuilding, omnis, and amp gem fire rate effects
Edit 4: added armored hull data, discharger
Edit 5: added hull damage, shields, and more tactical implications