OK, here is the whole script saved as droplet. Just drop your doc files on it.
doc to docx.app.zip (72.9 KB)
It does all of your 5 points with the following limitations:
(1) It accepts only doc files. You can’t drop folders or non-doc files.
(3) The conversion is done by MS Word. That means the script has no influence on the conversion process.
In the past I’ve already seen complicated doc documents that looked a bit different as docx documents. If you have such documents you have to visually control the result. But this an MS Word problem, and I think you won’t find any better docx converter than MS Word himself.