I don't think there is a system that satisfies your definition of uniformly weak mixing. Such a system must be ergodic, and therefore admit Rohlin towers. Given $N>0$ and a Rohlin tower $\{E,TE,\dots,T^{m-1}E\}$ with $N = o(m)$, you can set $A=B = E\cup TE \cup \cdots \cup T^{m/2}E$ and find that $\frac{1}{N+1}\sum_{k=1}^{N+1} \mu(T^{-k} A\cap A) \approx \mu(A)\approx 1/2$. In other words, given an initial segment of integers, every ergodic MPS on a nonatomic probability space admits a subset which is approximately invariant under that segment.

I should have read more carefully. I believe the method you describe was used in Frantzikinakis's article referenced above, and in Section 8 of the Ackelsberg, Bergelson, Best article.

My (incomplete) understanding is that the known proofs Szemerédi's Theorem admit no simplifications under the assumption $|A\cap [n]|\geq \delta n$ for a fixed $\delta<1$, such as $\delta=1/2$ in your question. Of course, if $|A\cap [n]|>(1-\frac{1}{k})n$, then $A$ contains $k$ consecutive elements (and therefore an AP of length $k$), but if you want the length of the AP to tend to $\infty$ with $n$, this doesn't help. Unfortunately I can't point to a specific reference discussing this fact.

If one is interested in detailed structure of topological dynamical systems, one could plausibly decide that nilsystems (i.e. translations on compact quotients of nilpotent Lie groups) form an interesting class, and that nilsequences are of special interest. One may develop the theory of 2-step nilsequences, as Host and Kra did here. I'm leaving this as a comment because I can't really imagine sustaining the effort only to understand a class of dynamical systems.

Theorem 1 of Pollington's paper proves that if $(r_n)_{n\in \mathbb N}$ is lacunary, the set of $z$ where $z^{r_n}$ is bounded away from 1 has positive Hausdorff dimension.

The union of the $A_i$ is all of $\mathbb R^2$, so for a given $S\subset \mathbb R^2$ we have $S=\bigcup_{i\leq 3} A_i\cap S$. If $\mu(S)>0$ then $\mu(A_i\cap S)>0$ for some $i$, as well. If $S$ is bounded and $\mu(A_i\cap S)>0$, then $A_i\cap S$ will also be a counterexample to the generalized density theorem.

I did assume the stronger version of the density theorem; I don’t immediately see a short proof of the stronger version from your hypothesis. You can get a bounded set from the example in this answer by taking the intersection of a bounded set of positive measure with the preimage of [i/3,(i+1)/3).

Gro-Tsen: Hindman and Strauss's book Algebra in the Stone-Čech Compactification is a comprehensive reference with detailed proofs of this and related facts. It's a fun read.