NO they are not the same. I am having the same problem. I bought a 2000 R6 that was used as a track bike and I am putting it back street. I have found out that the battery for the 99-00 is a YT12B battery. The 01+ use a YT9B. The main difference is the 12b is a 6w x 5 1/8h. The 9b is 6w and 4.x h.
My problem is I ordered fairings from a website and did not specify my bike was a 2000. (NEWB MISTAKE) They sent me 2001 fairings. I thought no biggie I will just change out the battery box and be done.
WRONG!!!! Well come to find out my fairings and the new battery box fit together great but now I have had to mod the battery box to fit CDI unit and now I cannot find a battery that will fit in the battery box that will work with my bike.
I have been getting all my parts from http://www.partforyamaha.com/
they have good prices but shipping is kind of high IMO.
So I guess to answer your question no the battery's are not the same between 99-02.
As for the tail light I too am having the same issue. Since I have the 2001 fairings and the double round taillight, my connector is a small rounded triangle looking connector. I am thinking about cutting off the connector on my 00 tail light and splicing it to the wires on the 01 tail light. But I would like to have my battery so I can test and see if it is going to work.
Sorry for the long post. But I hope it helps.