The link you have posted is to the RoboHelp for Word section. That was a separate program that used Word as the editor. It was discontinued several versions of RoboHelp back.
You can either paste into RoboHelp or import the documents. The latter is described on my site. See Importing and Linking Word Documents
If you want to paste in, the quickest way to clean the code is select all (CTRL A) in Design View and then click the Clear Formatting icon.
You will then need to work through the text to apply the styles from your CSS.
See www.grainge.org for free RoboHelp and Authoring information.
This is where the problem & the inefficiencies lie.
In my view MS Word is not a suitable for this process due to the formatting rework required 'post' import/paste but currently it seems to be the only option available to me outside of purchasing robohelp licenses for my 100+ authors. This is not an economically viable option & also would not be a desirable process solution for other specific reasons.
I would be surprised if this was not a common problem out there. My EMEA colleagues actually had to move off RoboHelp to get around this. Rightly or wrongly, it factored in their strategic decision.
Am I right in thinking that others work in Word and then you paste in?
What did the EMEA people use instead?
I have a similar situation to you where teams use Word to author disaster recovery documentation so I have direct experience with what you may be seeing. I link to those Word docs, which are stored in a network folder. I don't know whether you tried to use the 'linking to Word' functionality for this, but based upon what you wrote above, I would at least explore it. Peter's site has a lot of good information that can be used as a springboard to getting started with it.