I couldn't find anywhere that said one way or the other, do the usual restrictions of Wild Shape apply to Elemental Wild Shape?

Yes. Nothing about Elemental Wild Shape ability says otherwise. However,

If not, the Beast Spells feature allows a Druid to cast spells while shapeshifted, but specifically mentions beast forms. Can a Druid with the Archdruid feature cast spells while in elemental form?

The Archdruid feature has nothing to do with it, but yes, a druid with both Elemental Wild Shape and Beast Spells can cast spells in elemental form:

Beginning at 18th level, you can cast many of your druid spells in any shape you assume using Wild Shape.

The next sentence mentions "a beast shape", but like in the Wild Shape ability itself (which often refers to "beast form"), the Elemental Wild Shape ability is a special case that overrides this.

I couldn't find anywhere that said one way or the other, do the usual restrictions of Wild Shape apply to Elemental Wild Shape?

Yes. Nothing about the Elemental Wild Shape feature says otherwise. However,

If not, the Beast Spells feature allows a Druid to cast spells while shapeshifted, but specifically mentions beast forms. Can a Druid with the Archdruid feature cast spells while in elemental form?

The Archdruid feature has nothing to do with it, but yes, a druid with both the Elemental Wild Shape and Beast Spells features can cast spells in elemental form:

Beginning at 18th level, you can cast many of your druid spells in any shape you assume using Wild Shape.

The next sentence mentions "a beast shape", but like in the Wild Shape feature description itself (which often refers to "beast form"), the Elemental Wild Shape feature is a special case that overrides this.