I was going through the article, https://learn.microsoft.com/en-us/azure/architecture/patterns/cqrs which says, "If separate read and write databases are used, they must be kept in sync". One obvious benefit I can understand from having separate read replicas is that they can be scaled horizontally. However, I have some doubts:
All of these have "it depends"-like answers...
Yes, usually, although some implementations might choose to (try to) update read models transactionally with the update. With multiple nodes you're quickly forced to learn the CAP theorem, though, and so in many CQRS contexts, eventual consistency is just accepted as a feature, as the gains from tolerating it usually significantly outweigh the losses. I suspect the bit you quoted anyway refers to transactionally updating the write store with publishing the event. Even this can be difficult to achieve, and is one of the problems event sourcing seeks to solve.
Yes. It's trivially obvious - in this context - that data must be written before it can be read, but your apps as consumers of the data see them as read-only.
Both are valid outcomes. Usually this part is less an application concern and is more delegated to the capabilities of your chosen read-model infrastructure (Mongo, Cosmos, Dynamo, etc).