Mechanics on cooldown based weapons:
Cannons:
Cannons no longer have global cooldowns preventing them being fired together, they now work similiar to beams with accumulative power drains.

Torpedo Launchers:
All torpedo launchers have a 3 second global recharge time, meaning you have to fire them alternately every 3 seconds.
Firing any torpedo launcher will put any other torpedo launcher equipped into the 3 second global cooldown (WARNING: I have not tested aft/fore torpedo cooldown lock, each side may be seperate but I suspect it would take around 3 seconds for most ships to turn that far, but I am happy to hear any personal experiences with fore/aft torpedo set ups)

Each torpedo has its own personal cooldowns (Unlike cannons, the times listed on the weapons info pages are the actual times it takes for the torpedos to be ready to be fired again).
Photon torpedos are 6 seconds, 2 of them will give you a fire rate of 1 every 3 seconds, these are the highest DPS for slots used torpedos.

Note: A third torpedo of a different type can be used with 2 photons, if you fire it off in the place of one of the photons when its personal charge is ready you can add it in for a bonus spike of damage/special while maintaining the fire rate of 1 every 3 seconds.
Quantum and Plasma torpedos have 8 second personal cooldowns, 3 of them will give you a 1 shot every 3 seconds, quantums will give you a higher dps then photons when 3 are equiped and plasma will do more hull damage then photons if enemy has lots of shield/shield regen.
NOTE: The third quantum and plasma torpedo will not give as much gain as the second as it only cuts off 2 seconds rather than a full 3 from the cycle.

Transphasic and Chroniton have 10 seconds personal cooldowns, that means 4 of them will give you a rate of 1 shot every 3 seconds, transphasic will give you the highest DPS out of all the torpedos at that point due to its shield piercing bonus, Chroniton will slow the enemys movement.

Note:The 4th transphasic or chroniton launcher will not be as effective as the 2nd or 3rd, cutting only 1 second off the cycle time compared to the 3 seconds the other two will give.
Tricobolt are special in the fact that they are not effected by BOff skills and have a shared cooldown of 30 seconds, this means the fastest you will ever fire a tricobolt device is 1 every 30 seconds but the total cooldown is apparently 1 minute. This should not effect the other launchers though (beyond putting them into a 3 second global).

So in short:
Max fire rate of torpedos is 1 every 3 seconds.
2 photons will give you the max rate
3 quantum or plasma will give you the max rate
4 transphasic or chroniton will give you the max rate
2 tricobolt is the maximum you can use.
Mines:
Ok I have to come clean, I have done very little actual tests beyond cooldown tests on these, I cannot offer any wisdom as to what the effects/damage the mines are like or really the activation time. I would go along with the same effects as the torpedo variants (although probably of lesser strength). I will just state the facts of cooldown.

Mines have a 9 second global cooldown.
Photons have a 12 second personal, making 2 give you a 9 second deploy rate but only cutting off 3 seconds.
Quantums and Plasma have a 16 second cooldown, 2 will give you a 9 second deploy rate, with the second cutting off 7 seconds (much better!)
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Chroniton and Transphasic have a 20 second cooldown rate, 3 will give you a 9 second deploy rate, but the third will only cut 2 seconds off the cycle.
I am not aware of any tricobolt mines, I have not seen anything stated about their existance, I assume there is but they will work on the same 30 second shared cooldown for tricobolt weapons.
So in short:
Mines max rate of fire is a deployment every 9 seconds
Photons take 2 to hit this max, but with a high diminishing return for the second.
Quantums and Plasmas take 2 to hit this max, with a reasonable diminishing return for the second.
Chroniton and Transphasic take 3 to hit this max, with an extreme diminishing return for the 3rd one.
2 Tricobolt should be the max you can use.
Hope this helps.