Inline function is basically the same thing as a view or a CTE, except that it has parameters. If you look at the query plan you'll see that the logic from the function will be included in the query using it -- so no, you can't index it and SQL Server doesn't cache it's results as such, but of course the pages will be in buffer pool for future use.
I wouldn't say that each call to CTE is a repeated execution either, since SQL server can freely decide how to run the query, as long as the results are correct.
For multi statement UDF each of the calls (at least in versions up to 2014) are separate executions, as far as I know, every time, and not cached in the sense I assume you mean.