@Iskarioth
So the special government notice is made to pair 2 people that have both exceptional talent in a field. Thus, their child would be hella talented too. However, there is a chance that there will be a huge age gap between you and your partner cuz you're trying to create a really really talented child (there might not be anyone in your age group that is also exceptionally talented in the same field as you).
The normal government notice just pairs a couple so that their child would be above average, and to make sure that their age gap isn't that large (most of the time, same age I think).
The girl's probably really talented (not stated in what field tho) so that's why she got the special government notice. However, since there are some downsides of accepting this special notice so, you get to make a wish and they'll grant it if they can. The girl accepts in order to cure the boy's disease.
I'm pretty sure that the boy's disease isn't incurable. It's just that they don't have the technology needed to cure it (only a few cases exist in which it was completely cured). So, Anegawa-san (her advisor) creates the special government notice to cure people with the disease that Neji has (his wife also had it too. Read the chapter for more details).
Lol I tried answering all your questions but some of your questions confused me and that some of the answers to your questions hasn't been revealed yet in the manga.