header width on the last page of the chapter

Answer 1

I suspect your last heading box is being constructed after your text finishes making the multicol boxes, so you're out of the scope of your change. It goes back to the old value.

You'd probably do well to add the fancyhdr package and use it. I believe it's well-behaved in multicolumn.

Okay, so it's almost certainly the scope thing. You're doing the adjustwidth in your new multi environment. When your text runs out in the multi envirnment, you haven't filled the last page; headers aren't set up until the page is filled. So your mutlti environment finishes the box, you leave the scope, and THEN the page is finished and emitted. Using the old width.

Set the header width and parameters outside the environment.

Answer 2

I ran your test document. It seems to have a bug in that \newlength{\newtextwidth} has a global effect, and so causes an error. Not sure why that is but I pulled it out of the \newenvironment{multi} with no ill effect.

Charlie's diagnosis is definitely correct. An alternative solution is to end the page after the multicols but before the adjustwidth, thus:

\newenvironment{multi}[1]{%
   ...}
{\end{multicols}\vfill\break\end{adjustwidth}}

I have tested this solution and on your sample document, it produces good output.