Section 3.2 doesn't say that $^IW$ is an interval. It says it's a graded poset.

The interval in Bruhat order from the identity to $s_2s_1$ includes both $s_2$ and $s_1$. --- Having reread what you've written, it seems that you are using a nonstandard definition of "interval".

Nothing in your proof says that the map to the interval is surjective, and in fact it is not.

For the symmetric group on three letters, with simple roots $\alpha_1,\alpha_2$ and $I=\{\alpha_1\}$, we have that $^IW$ is not an interval in Bruhat order.

The way you have written the monomials in the question, they are of a particular form (eg, the first one starts with $x_1$, the second with $x_2$ and so on). If you just want to say "Let J be an ideal generated by a collection of degree $d$ monomials $m_1,\dots,m_r$", it would be clearer if you just said that. If you mean for the monomials to have some special form, I still don't know exactly what form you want. It is also confusing that you write "I think the value of $d$ for such a membership...", since $d$ is part of the given data.

The two different definitions for $J$ are not equivalent, and from what is written it sounds a bit more like you mean for $J$ to be any monomial ideal. Could you clarify?

A couple of words of caution: the longest word is typically not itself in $^IW$. Also, $^IW$ is not an interval in Bruhat order. Both of these facts are already visible in the symmetric group on three letters.

Since the conditions in question 1 say nothing about the relationship between $x_1,\dots x_4$ and $x_5,\dots,x_8$, the answer is "no". Same for Question 2. Some further conditions would be needed. (I thought of asking that the barycenters coincide, but that is still too weak.)

It seems like your definition would include a triangle with one edge removed (but with all three of its vertices). This seems kind of pathological (and in particular wouldn't be a CW complex). I am not sure how to rule out such behaviour by a purely combinatorial criterion, though. It seems like you are looking for a combinatorial abstraction of polyhedral complexes, and I am not sure there is any good one.