How can I use a counter with a dynamically-created name in Latex Expl3?

Internally, int-type expl3 variables use TeX \count registers, which must be defined before they're used. So something like

\int_use:c { __example_some_int }

will only work if you have previously ran

\int_new:c { __example_some_int }

The obvious solution here is to add \int_new:c to the definition of \__strxgen_construct_counter_name:nn, but that won't work, because \int_new:c is unexpandable, and you can only use expandable commands inside of a :c argument.

An alternate solution is to define a \__strxgen_construct_counter:nn that almost works like the current command, except instead of returning the target csname as a string, it defines a control sequence that points at the same underlying register. So whereas before you could use

\int_use:c { \__strxgen_construct_counter_name:nn { #1 } { #2 } }

you'll now need to use

\__strxgen_construct_counter:nn { #1 } { #2 }
\int_use:N \l__strx_last_counter_int

Complete implementation:

\documentclass{article}

\ExplSyntaxOn

\str_new:N \l__strx_counter_csname_str

\cs_new_protected:Npn \__strxgen_construct_counter:nn #1 #2
{
    \str_set:Nn \l__strx_counter_csname_str { g_strx#1_#2_count_int }
    \int_if_exist:cF { \l__strx_counter_csname_str } {
        \int_new:c { \l__strx_counter_csname_str }
    }
    \cs_set_eq:Nc \l__strx_last_counter_int { \l__strx_counter_csname_str }
}

% For user purposes
\NewDocumentCommand\GetValue { m m }
{
    \__strxgen_construct_counter:nn { #1 } { #2 }
    \int_use:N \l__strx_last_counter_int
}

\NewDocumentCommand\IncrementValue { m m }
{
    \__strxgen_construct_counter:nn { #1 } { #2 }
    \int_incr:N \l__strx_last_counter_int
}

\ExplSyntaxOff

\pagestyle{empty}
\begin{document}
    \verb|\GetValue{div}{leviv}|: \GetValue{div}{leviv}

    \verb|\IncrementValue{div}{leviv}|: \IncrementValue{div}{leviv}

    \verb|\GetValue{div}{leviv}|: \GetValue{div}{leviv}
\end{document}

You never defined \g_strxdiv_leviv_count_int, so your \csname...\endcsname command ends up with \relax and that's the reason for the error message.

If you use the counter name you actually allocated, it works.

\documentclass{article}

\ExplSyntaxOn

% declare counter
\int_new:N  \g_strxdiv_leveliv_division_count_int
% initialize counter
\int_gset:Nn \g_strxdiv_leveliv_division_count_int { 1 }

% Construct the counter name dynamically
\cs_new:Npn \__strxgen_construct_counter_name:nn 
#1% first variable part
#2% second variable part
{
    g_strx#1_#2_count_int
}

% For user purposes
\NewDocumentCommand\Test { m m }
{
    \int_use:c { \__strxgen_construct_counter_name:nn { #1 } { #2 } }
    % <other stuff should be placed here using the counter once "reconstructed">
}

\ExplSyntaxOff


\begin{document}

\Test{div}{leveliv_division}

\end{document}