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.

Your AC will be the better of the two (creature's or unarmored).

PHB pg. 67:

Your game statistics are replaced by the statistics of the beast, but you retain your alignment, personality, and Intelligence, Wisdom, and Charisma scores. You also retail all of your skill and saving throw proficiencies, in addition to gaining those of the creature.

Although this looks like you lose unarmored defense, it already adds exceptions. The last bullet of the description of Wild Shape states:

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.

This means that you can choose Unarmored Defense, meaning you can choose to use the better of the two Armor Classes.

There are no requirements about multiclass stats for anything other than multiclassing. The best reasoning as to why this is true is the fact that the multiclass rules state that the necessity for high attributes for multiclassing is because you are learning a new set of abilities quickly. You are expected to have learned your original class over the course of years, where multiclassing can, in many games, happen overnight.

Important note:

When you wild shape your physical stats do change, so remember that your AC from the monk's unarmored defense will adjust based on your new Dexterity modifier in the creature's form. This also means that a Barbarian/Druid would calculate their unarmored defense AC from "10+Dex Mod+Con Mod" based on their new stats in beast form. Additionally, a wild-shaped druid that has a buff increasing their Dexterity will increase their AC according to the increased Dex mod value.

http://dnd.wizards.com/articles/features/rules-answers-january-2016

Natural Armor: 10 + your Dexterity modifier + your natural armor bonus. This is a calculation method typically used only by monsters and NPCs, although it is also relevant to a druid or another character who assumes a form that has natural armor.

Additionally

It is also important to note that your mental stats (Int, Wis, Cha) remain the same. This means that you use your Wisdom modifier and the animal form's Dex modifier to calculate the monk's version of Unarmored Defense for a monk/druid.

This does not apply to the standard armor class calculations for the animal's natural armor calculation, only the Unarmored Defense armor class calculation.

So, for a Druid/Monk/Barbarian, you would have three possible armor class calculations available:

• 1) Natural Armor = 10 + (beast form's) Dex mod + (beast form's) natural armor.
• 2) Monk UD = 10 + (beast form's) Dex mod + (character's) Wis mod.
• 3) Barb UD = 10 + (beast form's) Dex mod + (beast form's) Con mod.

You would select the best of the three Armor Classes as your AC in each beast form.