Printing answers with the exam class on a separate page

Question

I've seen some questions asking for solutions to this problem in this site; I've had it myself. My code only works for printing the answer as a list with the same numbering as the questions; it doesn't put them on a any particular format nor it makes the grading table. I modified the code from this answer.

Answer 1

The gist of the mentioned answer is to collect all the answers in a vertical box to dump them at a latter stage, while using an answer environment to do it (quite an elegant solution, in my opinion). My additions are:

1. Printing the points obtained for the correct answer.
2. Automatically printing the question number on each answer.
3. Printing the correct answer for a multiple-choice question.
4. Dumping the answers on a titled new page automatically.

Here's the code with comments of my understanding of how each thing there works (please correct it if I got anything wrong):

\documentclass[addpoints]{exam}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%CODE TO PRINT ANSWERS ON A SEPARATE PAGE %%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\answers{Answer key}%DEFINE THE NAME OF THE ANSWER PAGE.
%%%%% SET THE BOX WHERE THE ANSWERS WILL BE STORED
\newbox\allanswers
\setbox\allanswers=\vbox{}
%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%ENVIRONMENT FOR ANSWERS
\newenvironment{answer}
{%
    \global\setbox\allanswers=\vbox\bgroup%BEGINS A VERTICAL BOX (1)
    \unvbox \allanswers\par%%%RECURSIVELY ADD NEW ANSWERS TO THE \vbox.
    \hspace{2em}\makebox[1pt][r]{\thequestion.\ }%%%%%%THIS PRINTS THE CURRENT QUESTION NUMBER AFTER ALL THE PREVIOUS QUESTOINS HAVE BEEN ADDED TO THE BOX; THE FIRST \mbox IS RIGHT-ALIGNED, SO IT PRINTS THE NUMBER ON THE CORRECT COLUMN. IF ABSENT, THE 1 OF A 13 WILL BE ON THE SAME COLUMN AS A 9.
    \mbox\bgroup%PUTS EVERYTHING INSIDE AN UNBREAKABLE BOX, FOR QUESTIONS ARRANGED ON COLUMNS, BECAUSE THE ANSWERS WILL BE THEN PRINTED THE SAME WAY; \bgroup BEGINS SUCH A BOX...(*)
}%
{%
    (\pointsofquestion{\value{question}} \points)%PRINTS THE POINTS OBTAINED, IN PARENTHESES; USEFUL WHEN GRADING WITH ONLY THE ANSWER KEY AT HAND.
    \egroup%(*)...\egroup ENDS IT.
    \medbreak%A BIT MORE OF SPACE BETWEEN ANSWER, SO YOU CAN USE \displaystyle IF NEEDED.
    \egroup%ENDS THE VERTICAL BOX (1)
}
%%%%%%%%% /ANSWERS

%%%%%%%%%%%%PRINT ALL ANSWERS ON A SEPARATE PAGE
\newcommand{\showallanswers}{%
    \par\pagebreak%NEW PAGE FOR OUR ANSWERS
    \pagestyle{empty}%
    \makebox[\linewidth][c]{\hfill{\Large\bfseries\answers}\hfill}\par%CENTRED BOLDFACE TITLE FOR THE ANSWER KEY.
    \null\par%EXTRA SPACE FOR THE TITLE, OTHERWISE IT LOOKS VERY CRAMMED.
    \unvbox\allanswers}
%%%%%%%%%%%%%%%%%%%%%%/PRINTANSWERS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%% TO-DO:
%1. Let the answer environment decide whether to print points; not print them if the question has no points.
%2. Use only the command \CorrectChoice to print the correct answer in a multiple-choice question, without the '\setcounter{correct}{\value{choice}}' line.
%DO ALL OF THE ABOVE REDEFINING THE solution ENVIRONMENT ALREADY DEFINED BY THE exam CLASS.
%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\begin{questions}
\pagestyle{empty}
\newcounter{correct} %COUNTER TO STORE THE VALUE OF A CORRECT CHOICE.
    \question[1] Did John Doe sign the Magna Carta?

    \begin{answer}
        He was busy baking bread, so he couldn't make it.
    \end{answer}

    \question[3] Who's Hamlet's true friend?
    \begin{answer}
        None, according to himself.
    \end{answer}

    \question[6] When was the Necronomicon written?
    \begin{answer}
        In the Age of Madness.
    \end{answer}

    \question[15] How many modes are there to skin a cat?
    \begin{choices}
        \choice 1
        \CorrectChoice\setcounter{correct}{\value{choice}} 20
        \choice 4
    \end{choices}
    \begin{answer}
        \Alph{correct}
    \end{answer}
    \question[4] What is \textbf{the} answer?
    \begin{choices}
        \CorrectChoice\setcounter{correct}{\value{choice}} 42.
        \choice 62.
        \choice 12.
        \choice 2. 
    \end{choices}
    \begin{answer}
        \Alph{correct}
    \end{answer}


\end{questions}


\showallanswers
\end{document}

As commented in the code, I would like to add the following features:

1. Let the answer environment decide whether to print points; not print them if the question has no points. I suppose this would require a \ifpoints command or something similar; I tried reading the source for the exam class, but I was quite lost.
2. Use only the command \CorrectChoice to print the correct answer in a multiple-choice question, without the \setcounter{correct}{\value{choice}} line.
3. Do all of the above redefining the solution environment already defined by the exam class. I tried it, but got too many errors and, again, got quite lost at reading the code for the class.