I don’t think so. Blockchain is an append-only log with consensus, and Nostr doesn’t need the consensus part because it’s not harmful if clients disagree about the state of the data. However, client data is signed with Nostr which is awesome, so you can easily tell whether something is legit. If a client or a relay misbehaves it’d be very easy to tell. I really love this architecture.

No. Some clients include the ability to make lightning transactions via them but that's not part of the protocol.

IMHO blockchain is an ugly terrible solution to a problem that had no other solution. The chain just gets bigger never smaller, requires expensive proof of work, requires complex synchronization protocols. Why so many people want to copy the worst part of bitcoin for their new billion dollar startup idea has always been beyond me.

No, nostr does not have a blockchain.

No, which is why it lacks incentives i.e. no economic reason to run relay nodes to store and transmit data.