- Dolphin is not a party into any of this. Valve's ToS likely allows them to take down anything for any reason they want. There's no counter claim process or anything like this.
- Valve could have decided to ignore Nintendo with ~ no liability. They decided to just do whatever they were asked, and that's not surprising given they initiated contact in the first place.
- Dolphin probably has no recourse here to get any other outcome from Valve, but also no particular risk or liability.
Now onto Nintendo's legal claims: nobody can tell for sure whether Dolphin is in the right, or whether Nintendo is in the right. Like all legal matters, there is a lot of space for interpretation.
Dolphin does distribute the Wii AES-128 Common Key which is used to encrypt Wii game discs. This isn't required in theory, the tools that dump game discs could just dump decrypted images, in fact that might be easier than dumping encrypted images (the decryption is done transparently by the Wii OS).