I've never used NiCd batteries, so I do not have any first hand experience however...
The differences between NiMH and NiCd are quite easy to find online.
NiCds are generally used in high drain applications where you need a lot of amperage out of your battery pack. I'm paraphrasing from a quick google search I just did now...
NiCds generally hold less power than a similarly sized NiMH. However, NiMHs self discharge quickly. They'll drain themselves if you don't use them. Generally you'd want to charge them right before you use them, or the night before. They DO have new NiMH batteries out that improve on this.
NiCds sometimes have a problem called the Memory Effect, which improper charging can cause. So, you have to be aware of that. Google it to find out more.
From what I'm reading online, NiCds can be charged and recharged more (but you have to worry about the memory affect). With that said, I bought NiMH batteries... probably close to 4 or 5 years ago and they're still going strong (although my charger doesn't seem to like to know when they are charged anymore, so I have to watch them closely).
In the end, I'd go with a NiMH unless you NEED a ton of current, which you probably don't unless you're planning on hooking up a ton of servos and they'll all be moving at the same time.
I can't help you with your battery charging black box power adapter. I'd suggest buying a real charger if you want the best life out of your batteries. Good chargers are relatively expensive though. Considering the voltage of a NiMH 6V battery pack would be closer to 7V when fully charged, I'm not sure how well that one you mentioned would work charging it. (Remember, a battery's voltage drops as it gets used. So it'll start out at 7V or so, then drop to... oh I dunno 4.5V or so when discharged. This is why we have voltage regulators). Then again, like I said before, I have no idea if it would work or not.