"async mode dedicated" is strictly for PPP and SLIP connections. "async mode interactive", on the other hand, can be used for PPP, SLIP, ARAP, along with EXEC access to the router. Suppose you're dialing into the router's AUX port to access the CLI. In this case you want interactive mode. If you're dialing into the router strictly for a PPP connection, use dedicated mode.
When using interactive mode, you can also use the command "autoselect" on the line to have the router automatically determine whether you want a PPP connection or an EXEC connection.
"async default routing" enables routing on an async interface by default. This means when you dial into the interface, routing is already set up. "async dynamic routing" means that the user must manually initiate the PPP session from the EXEC mode. "async dynamic routing" would be used if you have an "async mode interactive" for which you want EXEC access, and then want to call a PPP session.
Under normal use, you would pair "async mode dedicated" along with "async default routing" when running PPP over a dial-in connection. "async mode interactive" will be used to get remote access to the router via a modem attached to the AUX port. You most likely would not use "async dynamic routing", since you can just say "autoselect PPP" if you want interactive EXEC and PPP access.