How can the dimensions of a box be retrieved with LaTeX

You really shouldn't be referring to boxes by number even with the plain format. For example,

\newbox\mybox
\setbox\mybox=\hbox{Hello}
\showthe\wd\mybox
\bye

will work fine. Both \newbox and \newsavebox do a \chardef to assign the name you give as a number. So in LaTeX you can happily do

\newsavebox{\mybox}
\savebox{\mybox}{\parbox{4cm}{\lorem}}
\showthe\wd\mybox 

and all will be fine.

I believe the canonical LaTeX way is this:

\newlength{\myhh}
\newlength{\mydd}
\newlength{\myww}
\settoheight{\myhh}{\usebox{\Abox}}
\settodepth{\mydd}{\usebox{\Abox}}
\settowidth{\myww}{\usebox{\Abox}}