Here is an expl3 variant that puts the problems into "even" and "odd" sequences and then prints them using command \evenquestions and \oddquestions, respectively. I have put the answers into environments as from the OP they can be relatively complicated.
Here's the output:

You can of course put the odd and even numbered questions on separate pages by putting something like \newpage in between the \oddquestions and \evenquestions commands.
Here is the code:
\documentclass{article}
\usepackage{expl3}
\usepackage{environ}
\ExplSyntaxOn
\newcommand\Question[2]{%
\par\vspace{.25in}\hrule\vspace{0.5em}
\noindent\textbf{#1:}~\textit{#2}\par
\vspace{.5em}\hrule\vspace{.10in}
}
\seq_new:N \g_even_questions_seq
\seq_new:N \g_odd_questions_seq
\cs_new:Npn \add_to_sequence #1 #2 #3 #4 {
\seq_gput_right:Nn #1 { \Question{#2} {#3} }
\seq_gput_right:No #1 { #4 }
}
\NewEnviron{question}[2]{
\int_if_odd:nTF {#1}
{ \add_to_sequence \g_odd_questions_seq {#1}{#2}{\BODY}}
{ \add_to_sequence \g_even_questions_seq {#1}{#2}{\BODY}}
}
\newcommand\oddquestions{
\section{Odd~numbered~questions}
\seq_map_inline:Nn \g_odd_questions_seq {##1}
}
\newcommand\evenquestions{
\section{Even~numbered~questions}
\seq_map_inline:Nn \g_even_questions_seq {##1}
}
\ExplSyntaxOff
\begin{document}
\begin{question}{1}{Prompt 1}
Answer 1
\end{question}
\begin{question}{2}{Prompt 2}
Answer 2
\end{question}
\begin{question}{3}{Prompt 3}
Answer 3
\end{question}
% now print the odd and even questions
\oddquestions
%\newpage
\evenquestions
\end{document}
If all question numbers will appear (in the sense that if question k+1 appears then question k>0 also appears), then I would omit the first argument to the question environment and instead automatically insert it using a counter.
Edit
Rather using sequences I realised that it is better to use token lists. The following code does it this way, with identical output. For good measure, I have automated the question numbers:
\documentclass{article}
\usepackage{expl3}
\usepackage{environ}
\ExplSyntaxOn
\newcommand\Question[2]{%
\par\vspace{.25in}\hrule\vspace{0.5em}
\noindent\textbf{#1:}~\textit{#2}\par
\vspace{.5em}\hrule\vspace{.10in}
}
\int_new:N \g_question_int
\tl_new:N \g_even_questions_tl
\tl_new:N \g_odd_questions_tl
\cs_new:Npn \add_to_questions #1 #2 #3 {
\tl_gput_right:Nn #1 { \Question{\int_eval:n {\g_question_int}} {#2} }
\tl_gput_right:No #1 { #3 }
}
\cs_generate_variant:Nn \int_if_odd:nTF {NTF}
\NewEnviron{question}[1]{
\int_gincr:N \g_question_int
\int_if_odd:NTF \g_question_int
{ \add_to_questions \g_odd_questions_tl {#1} {\BODY} }
{ \add_to_questions \g_even_questions_tl {#1} {\BODY} }
}
\newcommand\oddquestions{
\section{Odd~numbered~questions}
\tl_use:N \g_odd_questions_tl
}
\newcommand\evenquestions{
\section{Even~numbered~questions}
\tl_use:N \g_even_questions_tl
}
\ExplSyntaxOff
\begin{document}
\begin{question}{Prompt 1}
Answer 1
\end{question}
\begin{question}{Prompt 2}
Answer 2
\end{question}
\begin{question}{Prompt 3}
Answer 3
\end{question}
% now print the odd and even questions
\oddquestions
%\newpage
\evenquestions
\end{document}
if clause(like if the pagenumber is even do this if not that this …). – current_user Sep 04 '18 at 17:10\verbor averbatimlike environment -- this includeslistings)? Would it be ok for you to use an environment for input (I guess that the answers should be moved, too)? – Skillmon Sep 04 '18 at 17:24