Egol has pretty much hit the nail on the head.
You have a big old mess there and you don't want to end up doing more damage than good - it is easily done!
Now the content is different enough that Google doesn't seem to notice that its the SAME FREEKIN THING on each domain, but I am very concerned. The company has their main multi-page domain which has other links and sources of traffic, but in essence the previous owner created a micro link web.
That there is a big problem - link networks, as that is what this basically is, are a huge headache and I can only assume it is because it is poor quality, that Google hasn't found it yet. However, they probably will at some point so you want to back out of that as soon as possible.
Am I right in thinking that we have to build a bridge before we burn a bridge?
Yes... And no. If you get this bit wrong and suddenly Google becomes aware of what is going on in the background, you will bring them down on you quicker than you can take the sites offline. I am guessing that 99.9% of these sites have no reason to be there? If that is the case, I would be looking at a structured takedown / decommission.
Is it worth fixing up some of those other domains to have original content to try and bolster what we already have?
Probably not. If the content is how you describe it, Google has probably already made their mind up about the site and trying to fix a site of poor quality, can sometimes be an uphill battle because you are starting at a negative score. That said, it is difficult to advise exactly on this. You want to get out of this whole link network though.
Would it be better to combine everything into one website, or try and have different domains represent different things. For example Envato.com is an umbrella website with 8 separate websites operating under the same roof using different domains.?
Trying to combine the 'fluff' content of 1000 sites into a primary domain could seal your fate. I would advise heavily against doing this. Perhaps there are some sites with decent content on them that could be used to make up a blog post, but you would have to be careful about checking to see if was indexed before placing it elsewhere.
Where do I begin? I feel like I have started this project numerous times. I know the keywords, I know where the duplicate content is, I know the structure of the main domain, I am getting the structure of the entire link web.
I wouldn't want to advise incorrectly on this, but I would guess you need to look to start a structured takedown of the network, but you must handle this carefully. Either that or distancing the main site by removing links from the network to the primary domain.
I hope this helps a little.
-Andy