I think one of my controllers is dying, as its battery seems to drain really fast. I let my dualshocks and mic hooked up all the time to my PS3 via their mini-usb cables. Is it bad to do so? Could it negatively affect the batteries inside the devices?

I think that is bad for all rechargable batteries.

pretty sure you should let it die then charge it to get the most out of the batteries. You can just buy a longer usb cable instead of getting another controller, I have one that is around 10ft and I don't have to worry about my battery.

Maybe we should ask the guys at Tested. This is a question that has haunted me for all of my years. I keep getting conflicting reports on the Internet. Some say it doesn't have any side effects whatsoever, others say it'll harm the battery in the long run.

I believe early generations of lithium-ion batteries were very specific in that it was best to discharge them fully before charging them. Battery tech has progressed enough so that this is not really a problem with today's devices as far as I'm aware - however most products recommend that you fully discharge and recharge the device once a year to maintain it's battery life.  
 
With respect to the original question... it is not ideal if your battery is never drained (though it becomes moot if one of your controllers already loses a charge rapidly, requiring constant USB connection), though I am unsure on how "damaging" leaving a perfectly fine battery continuously charging can be. I would assume it must do something over and above simply never discharging the device as most battery chargers have an auto-off safe function, though this maybe just to save power?

Eh, no? I always have it plugged in when it's on unless I play somewhere else which is rare, and I usually just keep my PS3 controller in the system when I'm away as well as long as I'm playing the PS3 at the time and not the 360. I don't recommend it though. I'm much more of a neat-freak sometimes, so I prefer having it wrapped up and set aside somewhere convenient.

This is a chemistry question, batteries work on chemical energies, without a long winded chemical explanation it is ALWAYS better to not leave them plugged in the entire time, and it is ALWAYS better to let them deplete completely before recharging. However the chemical reaction will ALWAYS degrade over time, those two things only make it last longer, it depends ont he reaction but you could literally extend the life of a battery a thousand times if you follow this principle.