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.

The water must fit within a 5 foot cube, per the first sentence of the spell description. it's not 125 cubic feet in any shape, it's a bounding box.

This only determines the water you affect with the spell. It does not say anything about the spell effects being limited to this 5-foot cube, only the water you can cast the spell upon. You can stretch it out into ribbons if you like, since that's a simple shape permitted by the second bullet.

You can't take it with you. “Animate” doesn't cover moving it (think of animation as just a succession of new shapings in-place), and there is an explicit limit of 5 feet in the first bullet for how far you can move the water.

It can't affect ice or snow, only the colloquial, common, non-chemistry meaning of “water” that means the liquid phase. You can see this colloquial meaning used in the last bullet where it talks about how you can “freeze the water”. (Wouldn't make much sense to substitute “ice” into that bullet point, would it?)