countersgroupinglengths
The
\setcounterand\addtocountercommands have global effect,
while the\setlengthand\addtolengthcommands obey the normal
scoping rules.
Why were they designed like this? Why are \setcounter and \addtocounter not defined to obey normal scoping rules?
AMS alignments are executed twice internally, a first pass to measure the fields and a second pass to typeset. \label is given a null definition the second time to avoid duplicating writes to the aux file which would generate errors.
So you do a global let of the real definition but you don't see it as you globally let to the null definition on the second pass.
You can see this by putting \show\mylabel inside your multline and see it reporting two values. You can use the \ifmeasuring@ switch to test which pass you are on. and just do the let on one of them.
\documentclass{minimal}
\usepackage{amsmath}
\begin{document}
\makeatletter
\begin{multline}
\ifmeasuring@
\global\let\mylabel\label
\show\mylabel
\fi
\end{multline}
\show\mylabel
\end{document}
Produces
> \mylabel=macro:
#1->\begingroup \measuring@false \label@in@display {#1}\endgroup .
l.14 \show\mylabel
There is a primitive \global that makes a following TeX assignment global. So you can use \global\setbox and \global\def = \gdef. In LaTeX as David Carlisle kindly points out you may get away with \global\setlength in some circumstances, but in general it is not guaranteed to work, and loading packages such as calc in fact prevents it working. One work around is to use an intermediate global TeX assignment of the result of a calculation via \global\len=\len after the \setlength:
\documentclass[a4paper]{article}
\usepackage{calc}
\newlength{\mylength}
\newlength{\myglength}
\newlength{\myllength}
\begin{document}
\setlength{\mylength}{10pt}
\setlength{\myglength}{10pt}
\setlength{\myllength}{10pt}
\verb+\mylength+ is \the\mylength.
\verb+\myglength+ is \the\myglength.
\verb+\myllength+ is \the\myllength.
\bigskip
\begin{minipage}{0.9\linewidth}
\setlength{\mylength}{20pt}
\setlength{\myglength}{20pt}
\global\setlength{\myllength}{20pt} % Doesn't work when calc is loaded
\global\myglength=\myglength
Minipage start
\verb+\mylength+ is \the\mylength.
\verb+\myglength+ is \the\myglength.
\verb+\myllength+ is \the\myllength.
Minigage end
\end{minipage}
\bigskip
\verb+\mylength+ is \the\mylength.
\verb+\myglength+ is \the\myglength.
\verb+\myllength+ is \the\myllength.
\end{document}
In the specific example you now give, you need to get hold of the totalheight of the box, to get a picture of the same height and this then needs to be shifted down by the depth of the minipage. The above \global tricks will now work:
\documentclass{article}
\usepackage{calc}
\usepackage{picture}
\usepackage{ragged2e}
\newsavebox{\questiontextbox}
\newsavebox{\questionpicturebox}
\newlength{\questiontexttotalheight}
\newlength{\questiontextdepth}
\newenvironment{questiontext}[1]%
{\begin{lrbox}{\questiontextbox}%
\begin{minipage}[t]{#1}}%
{\end{minipage}\end{lrbox}%
\noindent\usebox{\questiontextbox}%
\settototalheight{\questiontexttotalheight}{\usebox{\questiontextbox}}%
\settodepth{\questiontextdepth}{\usebox{\questiontextbox}}%
\global\questiontexttotalheight=\questiontexttotalheight
\global\questiontextdepth=\questiontextdepth}
\newenvironment{questionpicture}[1]%
{\begin{lrbox}{\questionpicturebox}%
\setlength{\unitlength}{1cm}%
\begin{picture}(#1-\fboxsep-\fboxsep-0.8pt,\questiontexttotalheight)}%
{\end{picture}\end{lrbox}%
\fbox{\raisebox{-\questiontextdepth}{\usebox{\questionpicturebox}}}}
\begin{document}
\begin{questiontext}{0.2\columnwidth}
\RaggedRight There must be some text here so I made it up to fill more than just
a single line.
\end{questiontext}%
\begin{questionpicture}{0.8\columnwidth}
\put(0.5,0.5){\circle{0.4}}
\end{questionpicture}
\end{document}
I have changed a couple of extra things in your code. Firstly, the width of your \fbox has not only the \fboxsep but two rules 0.4pt wide, so you need to substract 0.8pt more to avoid an overfull box. Secondly, the narrow column on the left is better set ragged right, using the ragged2e package provides the \RaggedRight command which is less drastic than standard \raggedright.
Best Answer
That is by design and coded accordantly in the LaTeX core source code. Counters in LaTeX usually count things which are independent from the local group. For example
figurecaptions are enclosed in the group added by thefigureenvironment and any change to thefigurecounter used by the caption would be lost afterwards. This would render the counter mostly useless.If you want to set a counter locally use
\c@<counter name>=value. To add something to it use\advance\c@<counter name> by value. To build the counter macro you can use\csname name\endcsnameor (mis-)use\value{name}which also works on the left hand side of an assignment.You can also define an own counter using the TeX macro
\newcount\yourcounterand then use\yourcounter=<value>etc. However, you then need to define the\theyourcountermacro yourself if you need it and can't use the normal counter formating macros like\arabic{..}or\roman{..}with it.