1: Is it necessary to perform a VACUUM and REINDEXING operation after restoring the dump from Postgres 9.6 to Postgres 13? The dump size could be 1/2 TB to 1 TB.
You can perform a VACUUM and an ANALYZE right after restoring, but you definitely shouldn't REINDEX.
2: Also, are there any other operations that are recommended to perform after pg_restore?
I don't think you need anything else.
3: What is the minimum required disk space if taking a dump on the same machine where the source database exists? Is it the "size of the current data folder x 2"?
There's definitely no rules like that. It's impossible to know before doing it.
