How do I get the screen with calculator etc to show without a Mac mouse?
The same way you do with a Mac mouse. Open the Calculator app from the Applications folder.
Or, you can add Calculator to the Dock to make it more handy.
How do you open other Applications? It's the same process to open the Calculator application.
How do I get the screen with calculator etc
Are you possibly referring to the Dashboard? It is also an App, but you can configure Mission Control to open the Dashboard using a mouse button. I don't use a Mouse, so I cannot bring up a picture of the Mission Control screen with Mouse options. You can set it to show the Dashboard with a keyboard key:
Open Mission Control System preferences:

Set Show Dashboard to a function key (with or without modifier keys), or to the right should be options to configure a mouse button press (if supported).