Is there any documentation on what is/isn't permitted on data.SE?
Specifically can I create:
Do I have my own schema to play with?
You're pretty much permitted to do anything that the system allows you to execute*. With that in mind, the database user used to execute user queries has the following set of permissions for any given site's database:
Excluding the operations from your list that require additional permissions leaves you with the ability to create temporary tables, since any user can do that.
No, you're running queries against the same schema in use by all other users of the system.
* Though running intentionally abusive queries (i.e. ones that make the server sad-face) is grounds for being banned, of course