random sampling DAGs via nilpotent matrix sampling
For question 2 the first answer is &quot;yes&quot;: if $M$ is a $0-1$-nilpotent matrix, then it is the adjacency matrix of some digraph.
Now $M^k_{i,j}$ is well-known to be the number of paths of length $k$ between vertices $i$ and $j$. Meanwhile, there exists some integer $\nu$ such that $M^k=0$ for $k \ge \nu$, so there are no paths of size greater than $\nu$ in the graph, in particular there cannot be any oriented cycle, hence the graph is a DAG.
For the rest of your questions, I have a mixed answer: indeed your going to get all adjacency matrices (as you're saying, this is just shuffling the topological order) ; and since the set of strictly upper triangular $0-1$-matrices of size $n$ along with the set of permutations of $n$ elements, each DAG of order $n$ will appear with positive probability.
Now I'm guessing what you want is that this probability is greater than some bound for any $n$, and this seems doubtful. Take for instance the path of size $n$, up to relabeling there are $n!$ of them. But there are $2^{(n-2)n \over 2}$  strictly upper triangular $0-1$-matrices of size $n$ and $n!$ way to shuffle each of them, so the probability of getting a path goes to $0$ exponentially fast.