AG groups don't sync user DB collation change to replica servers?
1) When we changed the user DB collation on server A, on server B, the user DB still shows the old collation. Is that by design? So we would need to redo AG to make the change show on the replica servers?


It may show the old collation, but it's been changed. Please note that changing the collation at the database level does not change it on any of the items underneath until it is specifically changed, however all new items will use that by default.

If you don't trust me (I'm a random internet person!) and you have a test system you'll see that this data is cached. Change the collation, then restart the secondary replica. Now check the collation. I do not know, though, why the metadata cache is not refreshed.


  2) When we change the columns and other objects' collation on server A, does that get changed automatically on the replica server through AG sync process? 


Yes, assuming it's a database level object in a database in the AG.


  3) Is there any advantage of having server collation being SQL_Latin1_General_CP1_CI_AI while the DBs being Latin1_General_CP1_CI_AI ?


Depends, does you application require a specific collation or options? The advantage is for the application to sort and display data properly.


  4) Lastly, if the default DB collation is different than SOME of the table/column collation, would that be an issue for AG/DB in general? 


It's not an issue for AGs, it's a potential issue for the application. The AG will happily send the log block changes all day.