From my experience, they never take your servers down without notice. Hetzner in particular has an abuse system where you need to submit a response within 24 hours after an abuse report. And even if you fail it will grant you a few more days to comply. Also in case you don't comply and you don't respond, it will just take down that server, not all your servers.
Abuse requests we had over the years are mostly related to copyright issues with certain images on client sites.
We also had a lot of DMCA requests sent directly to Hetzner by DMCA lawyers. But never had any issues with servers being taken down. We always responded in due time and if we couldn't fix a issue in 24 hours as they request, we just submit a response with a message saying we're in contact with the client to fix the issue and they always gave us more time to comply.
We also had cases of court orders to take down content, and entire articles defaming persons or celebrities, in which we protected the client who provided evidence of the case. Some were also news articles and had real reasons to be published under free-press laws and were also protected and not taken down despite the abuse reports to Hetzner.
So no, Hetzner will not take down your server without proper reason, also it will not take down an entire account if it is registered to a real company. Not even for not paying an invoice. It will disable the services after 30 days of not paying, not immediately, and even then it will wait a bit longer to fully remove the servers. Not sure after how much time server removal will occur as it never happened to us.
Also we had issues with DDOS attacks. In these cases, Hetzner will simply block an IP or re-route it to a null-address. In which case we just purchased another IP and now we have backup IPs on most servers with them. So we can simply just swap an IP with another in case of a DDOS attack and null-routing the old one.
