Mains transformer wiring
Yes, what you propose is correct. These transformers can be used selecting a serial wiring for 230 V AC or in parallel for 115 V AC.
Please make sure to connect the 115V connection to the 0 connection on the other winding.
You will remain with one free 0 and one free 115 V connection. The free connection can be wired to the live 230V and the 0 to the neutral.

The same principle is valid for two of the 9 V AC windings. Both parallel and serial connections are possible. If you need more current on the 9V side, then you can put the windings in parallel. If you need 18 V on the secondary side, then you can put them in series.

Note: I want to remind you that working with systems connected to mains voltages can be dangerous.