To progress from 1st Dan, the requirements are more or less the same as a color belt for my neck of the woods. Forms, sparring, board breaking, weapons. I think the difference between advancing in the color belt ranks and advancing in the black belt ranks is what the focus is on. At the color belt ranks we want to make sure that students are becoming proficent with all the basic material that color belts learn. The focus is proper technique, reaction force, body posture, etc.
At the black belt level, it seems from what black belt forms I know and the black belt forms I see that at our stage, each black belt level is supposed to heavily focus on one aspect of your technique and "take it to the next level".
For example, our 1st dans starting learning practicing single-arm techniques as proper class room movements. The reliance is speed and snap instead of reaction force. 1st dan form focuses on this change of how technique is used and you are required to stay at this rank for around as long as you were a color belt for, assuming you advanced quickly as a color belt.
Our 2nd dan form continues with the single arm movements, but has a lot more combinations with the hand techniques. It seems the idea is to develop more speed, better rythme, and better timing with the hands. Our kicks see a lot more combinations, but it looks more like they are trying to improve our balance on one leg.
The 3rd dan focuses heavily on kicks. I am still a 2nd dan waiting to test for 3rd, but after watching the form many times, it looks like by now they want to focus on the legs. At 4th dan our form has us executing techniques from the ground and kneeling positions.
Sorry for the long post. It was a long way to say it, but I think at the black belt level, instructors focus on taking what you have learned as a color belt and making it much better. Each rank should witness a rising quality of technique. So at 6th and 7th dan, I would say the difference is probably more leadership and knowledge. As for weapons; Stick, Nunchuck, bo, cane, and tri-sectional.