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.
You have the Con score of your beast form.
The first benefit of the Durable feat states (PHB, p. 166):
Increase your Constitution score by +1, to a maximum of 20.
This is a one-time benefit you receive when you take the feat. You level up, select Durable, and your Con score goes up by 1.
For contrast, take a look at the other half of Durable:
When you roll a Hit Die to regain hit points, the minimum number of hit points you regain from the roll equals twice your Constitution modifier (minimum of 2).
This is an ongoing benefit that is triggered whenever you roll a hit die to regain hit points.
So Durable increases your Con score by 1, but when you enter Wild Shape:
Your game statistics are replaced by the statistics of the beast.
and your Con score becomes that of the beast form.
This does open the game up to potential shenanigans around leveling up while Wild Shaped, but that issue already existed, and this is one of the least problematic examples.
Best Answer
Yeah, you get to keep it.
Check the SRD on wildshaping:
I've snipped some parts and added some emphasis, but you get to keep racial benefits if the new form can use them. That includes Fleet of Foot.