On the anonymous front, no, Nostr does not protect your digital identity by default, again, no app built with privacy in mind as it should be, your IP address is exposed by the clients logs and relayers if you are not using TOR Browser or a VPN. Amethyst app at least has in the settings the option of using TOR but you need to combine it with Orbot used as a VPN client on your mobile, so not for the non tech savvy.

Nostr is not fully decentralized in the sense that is not fully p2p, but relays in servers which act as relayers, some public, some private/paid (absurd IMO, since their selling point is censoring content considered spam or hateful etc so basically ironic and non sense considering the ethos of Nostr). But there are so many public relayers at the moment across the world that you can say is decentralized, but there is no current strong solution for true decentralization (no money incentive) nor for how it will scale further, currently is a tiny follow, we get to a serious usage/follow and the relayers will need a major upgrade