There are no rules that state that a wild shaped druid is not able to take or benefit from a short rest. There is a clearly defined list of what they cannot do in the PHB.
As there is no requirement to maintain concentration or focus on the shape, and because transforming does not break concentration on existing spells, a druid is able to meet the requirements for a short rest.
PHB Page 67
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.
PHB Page 186
A short rest is a period of downtime, at least 1 hour long, where the character does nothing more strenuous than eating, drinking, reading or tending to wounds.
Unless your form is described as a never stopping ball of energy, most forms are capable of resting, and therefore are capable of regaining uses of wild shape.
The phrase "gain the ability" means that it doesn't replace anything. In addition to your previous ability to wild shape as an action, you now also have the ability to wild shape as a bonus action.
If it had been "your wild shape ability now takes a bonus action, rather than a normal action" you'd lose the ability to wildshape as an action, but this is gaining a new ability, rather than modifying an old one. Nothing in that rules text has any effect on any other abilities you might already have.
Now, if you had somehow gained this ability without previously having the standard version of the wild shape ability, then you would be able to wild shape as bonus action but not a normal action... though at that point the rules might become pretty messy about what you could wild shape into.
And we have confirmation from Mike Mearls in this Sage Advice:
Bonus Action @BonusAction · Feb 17, 2016
.@SageAdviceDnD @mikemearls Can Druids with Combat Wild Shape still perform Wild Shape as an action, or ONLY as a bonus action?
Mike Mearls @mikemearls · Feb 17, 2016
@BonusAction @SageAdviceDnD you can do either - it doesn't replace the old option
Best Answer
This is the danger in using homebrew material. The creator does not typically have the benefit of an editorial staff or design team to truly vet the material before play. Also, the playtesting sample size is so small as to be almost non-existent. Because of this, homebrew material often requires on-the-fly modification or adjudication.
We can still apply some basic rules, although it would ultimately have to come down to DM adjudication. Let us proceed under the assumptions of the homebrew. There is nothing in the description of the Plant Wild Shape ability to suggest it works differently than ordinary Wild Shape upon the shape being brought to 0 HP. In the absence of specific, we must assume general.
When the gas spore drops to 0 HP two things must happen: 1) the gas spore explodes into spores. It is not only dead, it is gone. It has ceased to exist. 2) The druid reverts to humanoid form, taking whatever leftover damage might be involved.
If the gas spore disappears first, there is nothing left to turn back into a druid. If the gas spore reverts to druid form, there is nothing left that can explode. It's a binary choice.
Since the reversion to druid rule of Wild Shape normally circumvents any rules about death (the beast forms get no death saves and do not die), we should assume that the gas spore's explosion ability should come second. This is further reinforced by the fact that it would be irretrievable character death otherwise (no body), and no fun to play.
The most playable answer would be to have the spore revert to druid without triggering the explosion. The lesson here is to be very careful with homebrew and don't be surprised or disappointed if you run into problems when interacting with the official rules.