The strange example makes me confused - with + adverb : "Red items need dealing with immediately after the process..."
How correct and common of this structure?
The issue which makes it sound strange is not with + adverb; it's -ing + adverb. In this case, need behaves as a modal verb and should be followed by an infintive form.
To deal with (something) is a phrasal verb, and there's not really much you can do about the with. It's part of the verb and needs to be there.