If I order something from Saturn / Amazon for someone else's house, will they see who the invoice is going to on the package or similar, i.e. Who is the person ordering?

And maybe a second, stupid question: When I order a PS4 for the recipient, does the packaging give information about whether it is new or used?

I do not think so!

It depends on whether there's an invoice in the package or not, possibly your name as the buyer is in it. Only the recipient is on the package itself.

Warehouse items are marked as such.

An invoice is not included with Amazon, but with third-party providers (without FBA) it can be that something is included. More like a delivery note, invoices are less common.

As far as I can remember, Saturn always comes with a delivery note. Something like that looks like an invoice only without prices.