Are “this” and “next” demonstrative determiners?

Question 1: In the following, is this a demonstrative determiner:

I will go to the store this week.

Question 2: If so, then what class is next in the following:

I will go to the store next week.

Question 3: It seems that both serve the same grammatical syntax and function here; are they therefore classified the same? (If not, why not?)

Demonstrative determiner is not exactly a syntactic category; determiner is a syntactic category, but demonstrative is basically an etymological+semantic grouping consisting of this, that, these, and those.

So although next is a determiner in some cases, it is never a "demonstrative determiner".

As John Lawler notes in the comments:

This, that, these, those are demonstrative. Next is originally a locative (the superlative in nigh, near, next), but is frequently used metaphorically of time in sequence of events (The next speaker is Ms. Hobbs). Like all superlatives, and all demonstratives, it's definite, so there's lots of meaning overlap, but the demonstrative pronouns have their own paradigm and relate to other definite TH-words like the, thence, thither, thus, then, and there.

This is a self-answer. The various answers given don't seem to be complete. The best I can tell regarding the answer of this is "it's a bit complicated" and I refer the interested reader to look at the main comments for discussion. Particularly, John Lawler's comment:

You can't expect a great deal from syntactic category labels; they're rarely specific enough to be useful in every case.