How to deal with concrete security against lots of queries?
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.