Ask HN: checking/cleaning a domain's reputation with SEs

We purchased a domain from a competitor last year, because it was our brand with a different TLD and we were looking to grow our business in that country. The domain had been in use for more than 10 years by different companies and the seller may have engaged in shady or at least questionable practices (like offering "free counters" for web pages with links to that domain among others).

We have found a few backlinks from "adult" sites and got these removed, but no tools available to us give a complete assessment of links that might hurt us. Google Webmaster tools does not even display backlinks for this domain at the moment (even though it gets several 10.000 clicks from Google every day).

What could be done to assess the backlink status and how it might affect the domain's reputation with major search engines? Please recommend proven tools.