The absence of evidence is not the evidence of absence.
In this case, at least, the above is true. Since the wording for wild shape does not explicitly say conditions are removed/dispelled, and disease is a condition, that condition is not dispelled upon shifting back.
It can then be surmised that status affects such as poisoned, blinded, etc, are also not removed upon shifting to and from wildshape, it can be easily concluded that those effects still persist in the original druid's form.
This is not to say a disease is a standard condition like poisoned, blinded, etc. But it is still a condition that persists on the player until the effect wears off, or is cured.
Diseases specify what creatures are affected by them. For example, Cackle Fever:
This disease targets humanoids, although gnomes are strangely immune.
Or Sight Rot:
This painful infection causes bleeding from the eyes and eventually blinds the victim. A beast or humanoid that drinks water tainted by sight rot must... etc.
After this point, the real question becomes does a mundane disease that affects a specific animal affect the druid in their humanoid form?
That goes into some science that is more theory than actual rules. So there's no easy way to say that a disease will or will not affect the druid in a different form. However, it could be said that the disease remains dormant in an unaffected or immune host, but when the host becomes a viable creature the disease persists. This is up to your DM at this point.
- If the druid is considered their race in terms of biological susceptibility in wildshape or not, then the disease would always persist.
- If the druid is not considered their race in terms of biological susceptibility in wildshape, then the disease would not persist, or would be quickly cleansed from an unsupported host.
One last observation from this can be taken from the text of Wild Shape:
You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so.
Of course, biological susceptibility to a disease wouldn't exactly be considered a "benefit" but you could rule this to mean that if you are a Gnome you will, for the purposes of disease viability, be able to be infected as if you were a Gnome, no matter what beast you were taking the shape of.
To answer the first question: No. It would not carry over due to how Wild Shape works.
The reason: The only reason damage carries over is because you automatically revert when your HP is reduced to 0 in beast form. This means that damage is dealt after you revert technically.
However, if you revert as a result of dropping to 0 hit points, any excess damage carries over to your normal form.
Therefore the carry over of damage between forms is delayed damage to your caster form until you revert. No effect of healing causes you to revert to receive the extra restoration of hitpoints lost.
Exhaustion is a condition, wild shape says nothing about conditions, and thus, conditions remain when you change back and forth (if it was meant to allow you to shed conditions, it would say so, and also be even more powerful of an ability).
While HP and physical attributes do not transfer, conditions do (they are not explicitly excepted).