So is it possible to keep the unlock my computer with Apple Watch feature, but disable it for all other requests?
macOS doesn't provide any setting under the Security System Preference that lets you disable app authentication while keeping the setting to unlock the computer via your Apple Watch. It's either all in or out.
However, if you prefer not moving your hands away from the keyboard to Apple Watch or even Trackpad, you can still authenticate via keyboard. If you need a handy way to navigate through the on-screen controls, you can enable Use keyboard navigation to move focus between controls under System Preferences → Keyboard → Shortcuts.
Enabling this lets you use your keyboard to navigate between on-screen controls. Such as, in the following, you can use the Tab key to navigate to the Use Password... button and use either the Enter or Space key to press the button and enter the password, all without moving your hands away from the keyboard.
Similarly, is it possible to disable Touch ID on the MacBook Pro for unlocking things, e.g. when unlocking the secure system preference, but still keep it to unlock the computer from lock/sleep?
You can go to System Preferences → Touch ID and uncheck Use Touch ID for all the items except Unlocking your Mac.