Hi there,
read your question with great interest. This kinda thing is what I face during some projects to. The first problem is that you are the fourth company working on their site. Every SEO has his/hers own methodology. In other words, things you might not consider to do, might be considered or done by someone else. Then again, the basics should all be the same.
The fact they've had some bad SEO done in the past can hurt a site. I would disavow those links, wait for maybe a month or 2 and check regularly if the links have been dropped. And then look at what the rankings will do. But don't focus on rankings alone. For instance, what is the bounce rate at this point? If the incoming traffic is unrelated to the subject through directory links or bad links you might see a high bounce ratio or low views per visitor. That is also I factor I would look at after disavowing some links. In stead of disavowing everything at once (which you could do) maybe it's wiser to disavow 25% first, then another 25%, another 25% and then the last 25% of links. And keep measuring the effect.
As for using singular or plural domain names is always tricky. It also kinda depends on the language your focussing on. In the Netherlands we have some terms that the plural is completely something else as the singular version. Sometimes it's just a "s" on the end other times the entire word becomes something different.
I would work a bit more on the old domain, making sure the on page SEO is completely up to date and correct, making sure you disavow some/all of the links, creating new unique content, acquire new links through your new content and keep a focus on the future. That way you might also get this old domain to work. Besides the age of the domain could also be a factor (several tools take this into account).
If you do switch to another domain, I agree with Ray. Make sure you also host it on another IP adres and make sure you don't get any reference from the old to the new domain because then the bad link juice might just flow through to the new domain.
Hope my point of view helps
Regards
Jarno