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.

Raging while in Wild Shape

Wild Shape explicitly allows this. From PHB p 67:

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.

If a bear is physically incapable of flying into a primal Rage, then we are doing something very wrong as a society.

Using Wild Shape while in a rage

The only thing you cannot do while in a Rage is cast or concentrate on spells (PHB p 48). Although Wild Shape is a "magical transformation", it is never described as a spell and does not require concentration.

However, a Rage ends early if (PHB p 48)

...you haven't attacked a hostile creature since your last turn or taken damage since then.

This is a problem if you aren't Circle of the Moon (and what Bearbarian wouldn't be?), as vanilla Wild Shape takes an action, leaving you dependent on opportunity attacks or taking damage to sustain your rage.

Rage Damage with Natural Weapons

One of the features of Rage is a bonus to your damage while raging. This feature states that you can only apply if when making a melee weapon attack using Strength:

When you make a melee weapon attack using Strength, you gain a bonus to the damage roll ...

All melee weapons available to PCs are Strength-based, some have the finesse property. From PHB p 147:

...you use your choice of your Strength or Dexterity modifier for the attack and damage rolls. You must use the same modifier for both rolls.

Similarly, the damage bonus on natural weapon attacks is almost always equal to either the Strength or Dexterity ability modifier (there are exceptions for CR 0 creatures). Thus, it appears every melee natural weapon attack can be Strength based, although not necessarily using the published attack/damage bonuses.

Consider the Blood Hawk (MM p 319). Strength 6 (-2), Dexterity 14 (+2), its beak has an attack bonus of +4 and a damage bonus of +2. This is certainly consistent with its beak being a finesse weapon that is using its Dexterity score. A Raging Blood Hawk (what a great mascot name!) could use its beak as a Strength-based attack, with an attack bonus of +0 and a damage bonus of -2, and then add Rage Damage.

More generally, if your Strength modifier matches the published damage bonus, the attack is Strength-based. If the Dexterity modifier matches, it's a finesse attack, and you would need to recalculate the attack and damage modifiers bonuses using your Strength bonus to turn it into a Strength-based attack.