Indexes and Statistics on Temp tables
There can be benefit in creating indexes on temporary tables, but maybe not for a staging table. It's an "it depends" answer, unfortunately. You will need to test. If you posted the code for how you are interacting with the staging table, we could help determine if any indexes would help. An example of where an index might help is if you were joining the temp table to another table. If you were to index the joined column, there could be performance gains, especially if there are a lot of rows in the temp table.

You probably do not need to update statistics on the temporary tables. It's also an "it depends" answer, though I've never seen an update stats on temp tables in any of the thousands upon thousands of stored procedures I've looked at, nor have I needed to add it to resolve a performance issue.