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.
No.
A druid can only wildshape on his turn. So, while he can apply unlimited hitpoints to his form by wildshaping every turn, he cannot prevent you from dealing damage when it isn't his turn. If the druid is knocked out of his form (reduced to 0 hp), the first thing that happens is the excess damage carries over. But now he's essentially vulnerable until his turn. That's where the real damage comes in. You can think of wildshape as a regenerative shield of hit points.
Party Versus NPC Archdruids
There are a multitude of ways to "beat" the druid, just like any other encounter. Use your wits, bargain, stealth, combat, etc etc. Those tools are ALWAYS available. Your DM should be ready for you to avoid combat if you wish.
Party Versus PC Archdruids
If you're talking about PC vs PC combat, there are also a plethora of spells that can assist in killing the druid. For instance, if the druid has less than 100 HP in wildshape OR not, a Warlock can cast Power Word: Kill, and the druid simply dies. No save, just dead. That's just one spell. There are tons of others. A Druid can't wildshape if they're incapacitated. They, again, also can't wildshape unless it is their turn, so plan your heavy attacks accordingly. When a druid is knocked out of form, they are vulnerable, so maximize your damage in that time or make that vulnerability last longer with controlling spells. Any damage that exceeds a druid's wildshape form is applied to the druids regular form. Druids are very good at having health, so find a way to attack them in ways where health isnt the primary objective. Use conditions, spell effects, and any means to hinder the druid and he/she shouldn't be that tough of a fight.
Specialization
DnD 5e rewards specialization. An Archdruid specializes in wildshape to effectively gain temporary hitpoints among other combat abilities. It's the whole point of a wildshaping druid. Being a brown bear is cool, but when you can wildshape into one and get its free hitpoints, it becomes a big deal. That's largely what wildshaping druids are meant to do, so it only makes sense that it gets ridiculously good at it at level 20. All classes have a 20th level ability they get that helps them be even more specialized. It's a reward for forgoing multiclassing. Therefore, it should feel hard to deal damage to an Archdruid. But it's not impossible.
Best Answer
No
A class feature might give you a spell, but it explicitly says so in this case. See Barbarian's Path of the Totem Warrior for example:
A spell is a special term in 5e, a spell description always says its casting time, components, duration and range. For instance, only spells which duration is not Instantaneous can be dispelled.
The Wild Shape only says you assume the shape of a beast, not cast a spell:
That means it doesn't consume your spell slots, can't be counterspelled, and you can use it with your bonus actions on the same turn you cast a spell (providing your circle is the Circle of the Moon, otherwise you have to spend Action instead). It is still counts as a magical effect though, hence, wouldn't work in an antimagic field.