Elemental Wild Shape just grants access to different forms for you to Wild Shape into. As such, it's limited by all the restrictions and gains all the benefits of normal Wild Shape.
To answer the second half of this question, we need to know if Beast is a keyword, or if it's simply a general descriptor that could apply to anything we Wild Shape into. The answer to this is that yes, Beast is a keyword monster descriptor.
However, that is thankfully irrelevant. The text under Beast Spells refers primarily to Wild Shape, and has ancillary text that refers to Beast Shape. However, based on the text in Wild Shape, I believe that they are used in the text interchangeably and thus should be applied interchangeably for the rare case where you are not in a Beast form.
So yes, you can cast spells while an elemental using Beast Spell.
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".
Best Answer
An explicit exception would have to be made, but no such rule exists.
5e has a “specific beats general” rule, as seen in the intro the Player’s Handbook:
Since the feature description of Wild Shape stipulates, “You can’t cast spells”, a feature must explicitly create an exception to this to allow you to cast spells in Wild Shape, as the 18th level Druid feature Beast Spells does:
This is an exception to the previous rule for Wild Shape. Upon reaching 18th level, now you can cast spells while in Wild Shape. Which spells? Any spells? Spells known by the creature? No, the only spells this feature allows you to cast are your spells.
So to use a hypothetical beast’s innate spellcasting, you would have to also have a feature that explicitly creates an exception to the rule of Wild Shape that forbids spellcasting. To my knowledge, no such feature exists.
It should be noted that the Deep Rothe was updated in Monsters of the Multiverse; it no longer has the Innate Spellcasting trait, instead it simply has an action option that says:
Even under this change, it still doesn’t work for a Wild Shaped Druid because “you can’t cast spells”.
Glow-in-the-dark cows won’t break your game.
All things considered, allowing a Druid to cast dancing lights while Wild Shaped into a deep rothe probably isn’t going to break your game. The balance issue is giving the Druid full access to the arcane arsenal, something the 18th level feature doesn’t even do. It will be okay to let your Druid turn into a light-up cow.