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".

Yes, but only if they choose any of the Elemental Disciplines (or other class feature) that allows them to cast a spell.

In order to meet the requirement for Elemental Adept you must be able to cast at least one spell. That means that you have to actually currently be able to cast it.

Some elemental disciplines allow you to cast spells.

Thus, if you choose an elemental discipline that lets you cast a spell, then that absolutely counts for the requirement to be able to cast a spell. If you choose only disciplines that don't give you the ability to cast a spell (and you have nothing else that gives you this ability) then you do not qualify for the feat.

To be clear, simply being a Way of the Four Elements monk is not enough, you must have an ability from something that allows you to currently cast a spell. Choosing one of the elemental disciplines that grants you a spell is a great way to do this.

You'll know the disciplines that allow you to cast a spell because they will say some variation on:

You can [...] cast [spell name].

Currently, your friend does not qualify for the any feat that requires spellcasting

You say that your friend currently only has two elemental disciplines: Elemental Attunement and Fists of Unbroken Air. However, neither of these grants the ability to cast any spell, so they do not qualify for the feat currently.

Elemental Adept is useless without a spell to cast anyways

Spells you cast ignore resistance to acid damage. In addition, when you roll damage for a spell you cast that deals acid damage, you can treat any 1 on a damage die as a 2.

The feat's only effect is to boost damage from spells only. The feat would have literally no benefit if you couldn't cast a spell (that does that type of damage). This is, in fact, the reason that having the ability to cast a spell is a prerequisite for this feat.