Absolutely Not
As others have stated, no you can not use a focus (of any kind) while Wild Shaped, even if the animal you morph into has hands, and they are empty and holding the focus. All of the information you posted in the original question provides the answers for you so long as you cross reference them. But a lot of people have missed a crucial part of why it can't be done.
The description of the Druidic Focus says (5e PHB, p. 151):
A druidic focus might be [...] a totem object incorporating feathers,
fur, bones, and teeth from sacred animals. A druid can use such an
object as a spellcasting focus.
This essentially states that you get to create a focus out of whatever you deem worthy to call a focus; pretty neat.
The description of Material (M) components says:
Casting some spells requires particular objects, specified in
parentheses in the component entry. A character can use [...] a
spellcasting focus in place of the components specified for a
spell [but not costly or consumed components]
This is the start of the unravel. Here it says you can use a focus instead of a material component (so long as it has no cost/isn't consumed). So essentially focuses are just fluff/filler for material components that most people over look anyway, but if you don't overlook that stuff it's still a pretty minimal net gain.
Part of the description of the druid's Wild Shape feature says (5e PHB, p. 67):
- You choose whether your equipment falls to the ground in your
space, merges into your new form, or is worn by it. [...]
This portion infers that you get to wear or hold your focus if you so choose when you Wild Shape, which is pretty cool.
The druid's Beast Spells feature says (5e PHB, p. 67):
Beginning at 18th level, you can cast many of your druid spells in any
shape you assume using Wild Shape. You can perform the somatic and
verbal components of a druid spell while in a beast shape, but you
aren't able to provide material components.
This is the crucial spot. It specifically says no material components. If you recall from what exactly a focus is, it is essentially a fancy material component. Meaning it can't be used, not because you can't "hold" it, but because for Game Design purposes they disallowed it.
This is further backed up by the fact that they allow somatic components (hand and body gestures) even if you don't have hands or the appropriate body part to do the somatic component, i.e. a snake making a hand sign. So holding/wearing the material component/focus is irrelevant because it is completely outlawed in RAW.
The druid's Archdruid capstone feature says (emphasis mine):
At 20th level, you can use your Wild Shape an unlimited number of
times.
Additionally, you can ignore the verbal and somatic components of your
druid spells, as well as any material components that lack a cost
and aren't consumed by a spell. You gain this benefit in both your
natural shape and your beast shape from Wild Shape.
Doesn't this seem a bit redundant? It is because it is trying to show you that it is essentially an improvement to Beast Spells. At this point, once you reach 20th level you no longer need the material/focus component. So even at level 20 the answer is still technically no, but it becomes irrelevant because materials/focuses are no longer even needed.
However, this is D&D - so if the DM says, "Well that's stupid, you should be able to use your focus as long as you are wearing/holding it"... Then boom, there you go: the true answer is "yes, you can, so long as your DM handwaves it".
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.
Best Answer
Yep, Guardian of Nature will carry over and affect you while you're Wild Shaped. With some exceptions.
So, to start with, the spell won't end just because you use Wild Shape. Furthermore:
So the features you gain from the spell (which comes under "other sources") will work if the new form is capable of using them.
Anything that can walk should be capable of using this feature. I'd rule that a form which doesn't have a walking speed, such as a quipper or seahorse, isn't capable of using this.
As the quote says, you can't use darkvision unless your Wild Shape form already has it. If it does, it should get the 120 foot range. Probably. It's not completely clear.
Ok, this one's a bit weird, because monsters don't specify whether their attacks are Strength-based. You'll have to work with your DM on a case-by-case basis to find out whether this applies. (It's pretty easy to tell when a creature's attacks are Strength-based, though.)
Yep. Everything you can Wild Shape into has some sort of melee weapon attack, so you'll always be able to use this.