I just now found that issue on mine, I don't know if it is a bug or what. I think it is a bug. I went out of my job, opened a new one and divided the basic cube up then started to add hair, everything fine so far. Then, I went to Edit mode, moved one point on one hair and all the hair tools were disassociated with the hair and none of them would function with everything I had put in up to that point (luckily I learned to save my progressions as different named jobs). Also, whenever I had the UVmap disassociation error I noticed in the curves tab my UVMap wasn't correct so I reset it but, yours looks correct on that which leads me to think that any edit unhooks the hair sculpt tools.
Update: I started a fresh job, left the cube and duplicated it and pulled to the side. Added a few hairs on each and all sculpt tools work fine. Went to one hair on the first cude, edit mode for the hair, moved one point and sure enough, all sculpt tools for the hairs I put on were disconnected and unusable for the hairs on that cube. On the same cube I added fresh hairs just below the originals and those allowed me to comb and sculpt but the ones I had previously added and edited above them would not. In the object tree, it was still one set of curves yet the first ones (top rows) are not touchable and have no curve to manipulate like the ones below that hadn't been manipulated.
So, if anything is done to the curve, it quits being a hair curve apparently
Now I went back to yours and couldn't add hair...then noticed that your selection was set to points, I couldn't add any hair (I'm using 3.5.1).
I switched it to curves and it let me add hair.