It is not exact, Atiyah is wrong (or maybe at the time $\to 0$ did not mean exactness). Maybe he was thinking about 1-dimensional manifolds.

Then a locally free sheaf cannot belong to $Coh_{d,d-1}$ unless $d=dim X$.

I don't see anything missing or contradictory in what you wrote. A sheaf is (poly-)stable iff its reflexization is (poly-)stable. It would help, though, if you have explained what $Coh_{d,d-1}$ is.

Holomorphically (and the argument is very short and pretty, if I recall correctly)

Thanks! Yes, I was asking about an ifinite order element. I will make a correction

up to a constant, yes. Globally there could be a cohomological obstruction, think of $z\to \Re\log z$

Z is not a comples subvariety, an u is not closed; it is not har to find varieties which are not close, an the smallest complex subvariety containing them is the whole ambient variety; take, for instance, the graph of exponent in P2

I am not sure this could work; there is nothing prohibiting $Z'=M$, then $U'=M$ as well, your assumption $U=Z\cap U'$ will fail. however, it is not hard to find small-dimensional real analytic subvarieties which are Zariski dense.