Delimiters in the \seq_set_split:Nnn function

You have several errors in your code. Comments marked %%% refer either to the code at the left or to the code that follows

\documentclass{article}
%   --------------------
\usepackage[check-declarations]{expl3}
\usepackage{xparse}
\usepackage{amsmath}
%   --------------------
\ExplSyntaxOn
%   VARIABLE DECLARATIONS:
%%% the names are not compliant
\seq_new:N \Args_seq
\int_new:N \ItemCount_int
\int_new:N \index_i
\tl_new:N \aux_tl
\tl_new:N \sep_tl
%   --------------------
\cs_generate_variant:Nn \seq_set_split:Nnn {NVn}
%%% one of the arguments ought to be mandatory
\NewDocumentCommand\myMacroA{O{}O{,}}{\myMacroA_Main:nn {#1}{#2}}
%%% the name is not compliant
\cs_new:Npn \myMacroA_Main:nn #1 #2
  {
    \seq_clear:N \Args_seq
    \seq_set_split:Nnn {\Args_seq} #2 {#1}
    \myShowArgs{\Args_seq}
  }
\NewDocumentCommand\myMacroB{O{}O{,}}{\myMacroB_Main:nn {#1}{#2}}

%%% the name is not compliant; the function should be protected
\cs_new:Npn \myMacroB_Main:nn #1 #2
  {
    \seq_clear:N \Args_seq %%% useless
    \tl_set:Nn \sep_tl {#2}
    \seq_set_split:NVn \Args_seq \sep_tl {#1}  %does not accept the delimiter as a _tl variable.
    \myShowArgs{\Args_seq} %%% the call is not compliant
  }
%%% the name is not compliant and the code below can be shortened
\cs_new:Npn \myShowArgs #1
  {
    \tl_clear:N \aux_tl    
    \int_set:Nn \ItemCount_int {\seq_count:N #1}
    \int_set:Nn \index_i {1}
    \int_do_until:nNnn \index_i > \ItemCount_int
      {
        \tl_set:Nx \aux_tl {\seq_item:Nn #1 {\index_i}}
        \tl_use:N \aux_tl \text{~}
        \int_incr:N \index_i
      }
  }
\ExplSyntaxOff
\begin{document}
1. \verb+\myMacroA[a,b,c,1,2,3]+\\
\myMacroA[a,b,c,1,2,3]

2. \verb+\myMacroA[1;2;3;a;b;c][;]+\\
\myMacroA[1;2;3;a;b;c][;]

3. \verb+\myMacroB[a,b,c,1,2,3]+\\
\myMacroB[a,b,c,1,2,3]

4. \verb+\myMacroB[1;2;3;a;b;c][;]+\\
\myMacroB[1;2;3;a;b;c][;]
\end{document}

Now, let's look at compliant and simplified code. I used rn as your personal prefix.

1. The main argument to \myMacroA and \myMacroB ought to be mandatory

2. Function where variables are set should be protected.

3. If an argument to a function is supposed to be a variable, then the argument type should be N

4. With \seq_use:Nn the code for showing the sequence items can be simplified; if instead you want to do something to each item, use \seq_map_inline:Nn or \seq_map_function:NN

5. When you're using \seq_set_split:Nnn (or the variant) you don't need to clear the variable beforehand; the same for \tl_set:Nn

\documentclass{article}
%   --------------------
\usepackage[check-declarations]{expl3}
\usepackage{xparse}
\usepackage{amsmath}
%   --------------------
\ExplSyntaxOn
%   VARIABLE DECLARATIONS:
\seq_new:N \l_rn_args_seq
\tl_new:N \l_rn_sep_tl
%   --------------------
\cs_generate_variant:Nn \seq_set_split:Nnn {NVn}
\NewDocumentCommand\myMacroA{O{,}m}
  {
    \rn_MacroA_Main:nn {#1}{#2}
  }
\cs_new_protected:Npn \rn_MacroA_Main:nn #1 #2
  {
    \seq_set_split:Nnn \l_rn_args_seq {#1} {#2}
    \rn_ShowArgs:N \l_rn_args_seq
  }

\NewDocumentCommand\myMacroB{O{,}m}
  {
    \rn_MacroB_Main:nn {#1}{#2}
  }
\cs_new_protected:Npn \rn_MacroB_Main:nn #1 #2
  {
    \tl_set:Nn \l_rn_sep_tl {#1}
    \seq_set_split:NVn \l_rn_args_seq \l_rn_sep_tl {#2}
    \rn_ShowArgs:N \l_rn_args_seq
  }
\cs_new:Npn \rn_ShowArgs:N #1
  {
   \seq_use:Nn #1 { ~ }
  }
\ExplSyntaxOff

\begin{document}
1. \verb+\myMacroA{a,b,c,1,2,3}+\\
\myMacroA{a,b,c,1,2,3}

2. \verb+\myMacroA[;]{1;2;3;a;b;c}+\\
\myMacroA[;]{1;2;3;a;b;c}

3. \verb+\myMacroB{a,b,c,1,2,3}+\\
\myMacroB{a,b,c,1,2,3}

4. \verb+\myMacroB[;]{1;2;3;a;b;c}+\\
\myMacroB[;]{1;2;3;a;b;c}
\end{document}

You need a variant of \seq_set_split:Nnn that uses the value of \sep_tl, not the literal \sep_tl -- which is not the delimiter there.

With other words: \seq_set_split:NVn, which does not exist yet but can be generated with

\cs_generate_variant:Nn \seq_set_split:Nnn {NVn}

This \seq_set_split:NVn will now use the Value of \sep_tl (or any other variable that can be used with an V type.

Here is the shortened code, clearified, shortened, using the expl3 naming syntax, but first some comments:

• \seq_clear:N isn't necessary if \seq_set_split is used -- that command clears the sequence anyway
• \seq_use:Nn is much easier than glueing things together with some \tl_map_inline etc. loop.
• There are two scratch variables ('registers') for almost any expl3 datatype, that can be used for storing intermediate data, such as \l_tmpa_tl and \l_tmpb_tl or \l_tmpa_seq etc.

---

\documentclass{article}% Don't use minimal!
%   --------------------
\usepackage[check-declarations]{expl3}
\usepackage{xparse}
\usepackage{amsmath}
%   --------------------
\ExplSyntaxOn
%   VARIABLE DECLARATIONS:
\seq_new:N \l_neuwirth_Args_seq
\tl_new:N \l_neuwirth_sep_tl
%   --------------------

\cs_generate_variant:Nn \seq_set_split:Nnn {NVn}

\NewDocumentCommand\myMacroA{O{}O{,}}{\neuwirth_myMacroA:nn {#1}{#2}}
\NewDocumentCommand\myMacroB{O{}O{,}}{\neuwirth_myMacroB:nn {#1}{#2}}


\cs_new:Npn \neuwirth_myMacroA:nn #1 #2
  {
    %\seq_clear:N \l_neuwirth_Args_seq
    \seq_set_split:Nnn \l_neuwirth_Args_seq #2 {#1}
    \myShowArgs{\l_neuwirth_Args_seq}
  }

\cs_new:Npn \neuwirth_myMacroB:nn #1 #2
  {
    %\seq_clear:N \l_neuwirth_Args_seq% Not necessary since \seq_set_split clears the seq-variable
    \tl_set:Nn \l_neuwirth_sep_tl {#2}
    \seq_set_split:NVn \l_neuwirth_Args_seq \l_neuwirth_sep_tl {#1}
    \neuwirth_myShowArgs{\l_neuwirth_Args_seq}
  }

\cs_new:Npn \neuwirth_myShowArgs #1 {%
  \seq_use:Nn #1 {\c_space_token}
}

\ExplSyntaxOff
\begin{document}
1. \verb+\myMacroA[a,b,c,1,2,3]+\\
\myMacroA[a,b,c,1,2,3]

2. \verb+\myMacroA[1;2;3;a;b;c][;]+\\
\myMacroA[1;2;3;a;b;c][;]

3. \verb+\myMacroB[a,b,c,1,2,3]+\\
\myMacroB[a,b,c,1,2,3]

4. \verb+\myMacroB[1;2;3;a;b;c][;]+\\
\myMacroB[1;2;3;a;b;c][;]
\end{document}