While hinata&neji's element does seem to come out of nowhere, that does not mean they can't have different elements. I mean they are just cousins, whereas sasuke and itachi are brothers and yet one is lightning the other is fire. Blood relation does not mean exact same element.

At the same time, Earth is on neji's profile on the wiki while lightning on hinatas while neji do not have lightning and hinata does not have earth. Both are associated with more than one element, so I'm not sure how it is selected out of the "pool".

I do agree on the danzo one. He C*e fire, but he is mostly wind. i wonder if it's because his sharigan is quite prominently featured and the fact that most uchihas are fire?

According to wiki, Hinata is fire&lightning while neji is fire, earth and water.

Water needle is anime only, so depending on who you ask, might not be considered canon.

And as far as medical jutsu goes, hyuga clan's "gift" can be best compared to a microscope/magnifying glass that a surgeon have. Whereas water element is the scalpel etc tools that are used to perform medical procedure. So given two characters who are both equal water users, a hyuga would have an advantage. However, that doesn't mean every, or even most hyuga is a surgeon, that's backward logic--lots of people have microscope and are not surgeons. While given a choice, they might choose to learn water over other elements since they do have that advantage. But like you said, element is often associated with blood-line and is something people are, at least in part, born with. So it is perfectly possible that most hyuga cannot take advantage of their... advantage in medical jutsu because they don't have the more essential part of the gift.
Of course, eugenics could be a consideration, that is to say purposely marrying clan members to water users in hope of having a byakugan wielding water element child that's a perfect medical ninja. But there does not appears to be any hint at this particular eugenics practice being done.

It is true, however, that most hyuga is not associated with elemental release. Thou as the game have to give every ninja an element, they might get assigned to whatever they might be closer to. And in this case, for hyugas that does NOT have an elemental release, perhaps they would be assigned water or earth (earth due to its tie to taijutsu styles)

But like I said earlier, neji and hinata DOES have nature releases, so their element would come from that.