There are a lot of commands for quickfix as you have said, but I tend to find I only use a small subset of them:
:copen " Open the quickfix window
:ccl " Close it
:cw " Open it if there are "errors", close it otherwise (some people prefer this)
:cn " Go to the next error in the window
:cp " Go to the previous error in the window
:cnf " Go to the first error in the next file
:.cc " Go to error under cursor (if cursor is in quickfix window)
I tend to use this with :make and :vimgrep, so I can’t comment on the Javascript lint checker, but this should give you something to get started.
Regarding the general use of JavascriptLint, I’m not a javascript programmer, but it looks like the script exposes a function called “JavascriptLint”, so if you want to call it manually, you can use :call JavascriptLint(). However, it works on the disk copy of the file, so it’ll have to be saved first. If (and only if) the command line jsl works on html files, you should be able to use :call JavascriptLint() on an html file to check the internal javascript. You could also do:
autocmd BufWritePost,FileWritePost *.html call JavascriptLint()
to automate it. If jsl doesn’t support html files, then (short of patching the application or asking the author to change it), it’s probably a lost cause…