New elasticsearch node does not run enrichments

I'm not sure about the internals, but the indexing request to write on the shard is done after the ingest process, so the data is processed by the ingest pipeline, which will run the enrich processors and after that they will be written on the specific shards.

No, if you do not have a persistent volume for the node02, the data will be lost if you restart it, every data node needs to have persistent volume.

Do not restart your node02 if you have data on it and do not have a persistent volume mounted, you need first to remove the data from the node, remove the node and add a persistent volume for it to work.