_Could a specific url of my site be banned by google? _Yes. And like you said, its likely offpage (backlinks with overoptimized anchor text maybe?)
1. Is this a Google penalty? If you were ranking well and it dropped, something must have changed. Either algorithm or page on/offsite seo
2. Do you think it could be a SEO offpage, onpage or both issue? Could be either or both
3. Could it be that someone is sabotaging our project? If so, what is the best tool to find out (if it exists) check GSC for backlinks to your page
4. Any other good tool to evaluate on onpage? make sure the page isn't overoptimized for any terms (i.e. don't keyword stuff). Also make sure robots can reach your page (not blocked via robots.txt or meta robots)
5. Even if I do not get a notification on the Google Search Console, Can we ask to Google for a reconsideration? No. Reconsideration is ONLY if you received a manual penalty.. for which you would receive a message in GSC.
You gave a lot of info but this seems like a really specific problem and there are many variables at play. Hard to judge without knowing the specific pages.