> I can not believe this is the normal legal process.
It's not.
> There must have been some extraordinary circumstances that I am not aware of.
Possible, but so far, the case is mostly sealed (refer to the above statement). From what we know, Microsoft identified some botnet(s) using noip's service as part of their infrastructure. Nothing points to noip knowing about the botnet, so we must assume until we know otherwise that they had no knowledge. This leaves us with a US company waking up one day to find their property seized by another US company, supposedly with the backing of a US court.
> Given that Microsoft probably hasn't got more legal rights than anyone else, could No-IP do the same to Microsoft?
Probably not -- Microsoft has a history of helping Law Enforcement track down botnets -- in the past they have literally marched into data centers and seized all the servers from a company without their knowledge (they even did this overseas outside of the US once). It's unlikely if noip were to bring suit against Microsoft, that it would go anywhere.
(It's as scary as it sounds when a private company is leading Law Enforcement, instead of the other way around).