How can I create framed boxes with two columns, where there is a line numbering on every row where there is something written? (if hard or possible then a number for every line in general)
Something like this:

How can I create framed boxes with two columns, where there is a line numbering on every row where there is something written? (if hard or possible then a number for every line in general)
Something like this:

It is a lot easier to produce the corrected image! Please do bear this in mind when posting future questions. (I came up with a solution for the original even though it made no sense to me to do it that way!)
Boxes are no problem. The page I directed you to includes information about typesetting proofs with boxed sub-proofs in at least three different styles. You should try this one:
\documentclass{article}
\usepackage{logicproof}
\begin{document}
\begin{logicproof}{1}
P \lor Q & premise\\
Z & premise\\
\begin{subproof}
P & [assume]\\
[\dots] &\\
&
\end{subproof}
\begin{subproof}
Q & [assume]\\
[\dots] &\\
&
\end{subproof}
(P \lor Q) \land Z & el 3--4 5--6
\end{logicproof}
\end{document}

Incidentally, you ought not need sub-proofs at all for this particular question, but you know that, right?
tikzmark library to mark points in the proof and the fit library to construct a TiKZ node around the proof, overlaying the picture on top of the proof itself. But this is a bit fiddly ;).
– cfr
Nov 28 '14 at 00:15
tabularenvironment – Werner Nov 27 '14 at 23:29