The same intersection-theoretic argument seems to give the following bound: if $i\leq m-1$ and $j\leq n-1$ are such that $\binom{i+j}{i}$ is odd, then any subspace of codimension $i+j$ must have a pure tensor. The largest possible value of $i+j$ (for fixed values of $m$ and $n$) one can achieve this way is the bitwise OR of $m-1$ and $n-1$ (i.e., the number whose binary expansion has a $1$ in any given binary place iff either $m-1$ or $n-1$ (or both) do.)

The gcd condition is not nearly enough to make $m$ finite: let $m_n$ be the number of positive integers up to $a_1+\cdots+a_n$ which cannot be expressed as a sum of the $a_i$. (BTW, one has $A_n=a_1+\cdots+a_n+1-m_n.$) If $a_{n+1}=1+a_1+\cdots+a_n-i_n$ for some $i_n\geq 0$, then we must have $m_{n+1}\geq 2m_n-i.$ We can then choose a sequence with a large value of $a_2$ and $i_n$ equal to $0$ or $1$ for $n\geq 2$. If $i_n$ does not become eventually $0$ then this will satisfy the gcd condition but $m_n$ will blow up, which forces $m$ to be infinite.

I think this is very common, here's just one example off the top of my head: en.wikipedia.org/wiki/Category_of_relations.

@Luftbahnfahrer Yep, you are of course correct on all counts. This argument is incomplete as is.

I believe Milnor's "On the Betti Numbers of Real Varieties" gives you an upper bound $N(n,d)\leq d(2d-1)^{n-1}.$ Probably significantly stronger bounds are known today, but I'm not well-versed in the relevant literature.

There are only two "non-repeating" geodesics on the sphere between two different non-diametrically opposed points, corresponding to the two sides of the unique great circle through them.

Another example, using the same idea of taking a non-complete manifold where you delete points to force intersecting geodesics: take $M$ to be $S^2$ and take $4$ random points $A,B,C,D$ on it. For any bijection, there will be a pair of geodesics that intersect (since any two great circles intersect.) Delete points to ensure that those are the only geodesics left.

Yes, but by definition, your Q2 action is equivalent to the action of the skyscraper in $\operatorname{IndCoh_{nilp}}$ on the automorphic category (in particular it does not match up with Q1 at non-smooth points.) I.e, both questions concern endofunctors of $D(Bun_G)$ so it doesn't make sense to say they're interchanged by Langlands, they are just either identifiable or not identifiable.

Ah, so the difference between Q2 and Q1 is the difference between IndCoh and QCoh? Then I don't know a good answer (except when $\sigma$ is a smooth point and the question coincides with Q1), even in the case of $\sigma$ trivial (in which case I guess you could theoretically derive an answer from a description of the convolution kernel but I never understood the proposals for this kernel.)