How to deal with concrete security against lots of queries? In general, $O(1/\epsilon^2)$ queries are required to distinguish between two distributions that are statistically close at most $\epsilon$.
This informal state deals with the required number of queries to distinguish between two distributions. However, I cannot find any statement that when we have $Q$ queries, then we can distinguish two distributions under some $P$ probability.
I have a few questions about some common knowledge in cryptographic fields.

How can I formally write the statement about concrete security against lots of queries?
How we set parameter to be secure against adversaries allowed $Q$ queries? As standard decision problems, adversary has a negligible advantage?
Why NIST PQC standardization recommend parameters to be secure against number of queries $2^{64}$? Is there a special reason? What is implicit consensus relate to $2^{64}$?
Distinguishing distributions is studied in the field of distribution testing. See this survey for more information.
In particular, I believe the survey states that the correct sample bound includes a term roughly of the form $O(|\Omega|/\epsilon^2)$, where $\Omega$ is the support of the underlying distribution.
As for your particular questions:

Generally, query-bounds arise within cryptography when making Renyi-divergence arguments, as one can often make significant gains by bounding the queries (say as $q \leq 2^{64}$, rather than $2^{128}$, which is the &quot;trivial&quot; bound for 128-bit security). See for example this slides of Thomas Prest's (or in general his recent publications --- he has been quite interested in developing renyi divergence arguments in lattice cryptography).

For setting parameters when you're allowed to institute query bounds, see the slides. In particular, he works through the example of a particular algorithm (the Micciancio Walter convolution sampler) that claimed 100 bits of security, and shows that (up to $2^{96}$ queries) the sampler actually achieves 256 bits of security. For a different algorithm, he is able to set parameters such that certain state required for the algorithm is an order of magnitude smaller.


I can't really speak to the third question in any concrete way though.