You can do it for every open affine and then glue, because the isomorphisms are canonical.

The problem with taking a crepant resolution and then normalization is that the result will typically be singular.

Let me take a look at the case $n=2$ to get an idea of what one might expect. In this case the triangle $tT$ in question has vertices $$(td_1,0),(0,td_2),(0,0).$$ Pick's formula gives the number of points as $${\rm Area}+\frac 12\sharp({\rm pts ~on~the~boundary})+1 =\frac 12t^2 d_1d_2 + \frac 12t(\gcd(d_1,d_2)+d_1+d_2)+1 $$ so the coefficients are all positive.

I have heard that such successful arguments occasionally happen in experimental sciences, but I have not heard of one in math.

As far as (4) goes, this is probably the most annoying situation. However, the editor is typically not in a position to resolve scientific disputes and has to go with the referee report. So it is not surprising that appeals to the editor fall on deaf ears.

Regarding (2): this is a decision of an editor, which is likely rather uninformed and is based on a superficial reading of the paper. I have made such decisions and must have been wrong a certain percentage of the time. However, it just needs to happen if the volume of submissions far exceeds the journal's capacity. Hopefully, such a backlog rejection happens quickly, so a paper can be sent to another journal.

Step one: talk to Davesh Maulik :)

In my paper, I only consider $|t|<1$. The products would diverge, badly, if $|t|>1$. So the product formula is only valid in this regime.

Out of curiosity, would there be a nice formula for counting abelian groups $G$ with coefficient $1/|{\rm Aut}(G)|$?

Not sure what the problem is: you have poles at $y^{-1}=t^{-n}$ on one side and at $y=(t^{-1})^{-n}$ on the other side, which is the same thing (as viewed as a function of $y$, in appropriate ranges for $|x|$ and $|t|$).

Yes, you are right.

I would try to look at the subring of $C[x,y]$ generated by monomials $x^my^n$ with $m\leq n\sqrt 2$ or something like that.