FCP does not always remember settings for projects?

If you hover over icons, a popup will show you what the keyboard shortcuts are. For example, hovering over the Snapping icon shows that 'n' is the keyboard shortcut to enable or disable.

There are quite a few single key shortcuts. Here's a list of all shortcuts: Keyboard shortcuts in Final Cut Pro for Mac - Apple Support

These settings are not per project; they are overall in the application.

Furthermore, they have keyboard shortcuts, and it is very easy to press these keys unwittingly.

For example the letter s toggles skimming. If you accidentally press it when not editing any text field, boom, you turned skimming on or off.

I would not expect these sliders to move without user intervention.

As always when there is unexpected behavior, you can try deleting preferences. To delete preferences, hold down command and option while starting the application and click Delete Custom Settings. The application will start with default settings and open an "Untitled" library. Adjust the settings, reopen the library you were working, and see how it works now.