I downloaded this Family OS and I get "you're ineligible for claiming the install bonus" when I try to claim it in the rewards page. What should I do to be eligible?

How is the ownership of this corporation enforced via the cryptocurrency?
If I'm understanding correctly clients will only sync the corporation chain but not the main chain? Will it be possible to sync with all chains?
Is the mainnet ready?

Then IMO utxos being looked up from the same IP would be the only realistic concern (even then I don't think there's been a single proven case of this being exploited). You should be fine chainanalysis-wise on the mixed end if that's your concern. But remember that the fact of you having participated in the coinjoin may be easy to figure out. The government may not know what you did with your mixed coins, but they can audit you just for mixing coins and try to find something else.

Very specific to the implementation of your client. If new addresses are generated from a common seed then you need to make sure the xpub never leaks. How does your client looks up the utxos? What if your computer is compromised? Does the software leave any trace on your computer? Unencrypted logs etc? Only one things needs to go wrong. Heck, even if the implementation is perfect, a simple malware attack could link your addresses.