Impronving your google skills seems to be one way to go. Try different combinations and using the "-" prefix to unmatch words. In your case "cluster" often refers to huge EU research grants, so you may try to include "-EU" in your search.

A very similar construction appears in Hochster's "Some applications of the Frobenius in characteristic 0" in the examples in Section 3.

Ordinary Segre products are long known. According to Wikipedia Segre's 1883 thesis was on quadrics in projective space. I assume that is how the name came about. I'm specifically looking for the multigraded analogue where the grading is by an affine semigroup. (A vector configuration $A = \{a_1,\dots,a_n\}$ is a finite subset of $\mathbb{N}^d$, defining the semigroup $\mathbb{N}A := \{\sum_i n_i a_i : n_i \in \mathbb{N}\}$)

A relation between $\text{depth} I$ and $\text{depth} I \cap K[x_1,\dots,x_r]$ can hardly exist. If $r=n/2$, say, then $I=(x_1,\dots,x_r)$ and $I' = (x_{r+1}, \dots, x_n)$ are two ideals, both containing a regular sequence of length $r$. In the first case the entire regular sequence survives, in the second case nothing of it survives.

@Timothy, I think the most opaque part of this is why the intersection of the minimal primes comes out binomial in the first place. For me it has always been useful to think about the cellular case: An ideal is cellular if in the quotient every monomial is nilpotent or regular. Cellular decompositions into binomial ideals exist in polynomial rings over every field and in characteristic zero a radical cellular ideal is just a lattice ideal + variables. (Note that in characteristic zero lattice ideals themselves radical.)