What is the shortcut to start and stop recording?

Shift + Command + R by default. It can be changed in Settings under Shortcuts.

How do I split a clip with the keyboard?

Press S with the playhead where the cut should be. Backspace then deletes the selected clip or zoom.

Yes. Press Z in the editor to add a zoom at the playhead.

Reference

Every key the app answers to.

These work while any application is focused.

Keys	Does
Shift + Command + R	Start a recording, or stop the one running. Changeable in Settings.
Shift + Command + P	Pause or resume the recording.

Keys	Does
Esc	Cancel choosing a source, or cancel the countdown.

Ignored while you are typing in a field.

Keys	Does
Space	Play or pause.
S	Split the clip at the playhead.
Z	Add a zoom at the playhead.
Backspace or Delete	Delete the selected zoom or clip.
Command + Z	Undo.
Command + scroll, or pinch	Zoom the timeline.
Esc	Clear the selection in the composition, or close a dialog.
Option while dragging	Pan the picture inside its box instead of moving the box.
Shift while resizing	Keep the proportions.

Keys	Does
Backspace over selected words	Cut that footage from the recording.
Command + Z	Undo.
Esc	Leave the text.

What is the shortcut to start and stop recording?: Shift + Command + R by default. It can be changed in Settings under Shortcuts.
How do I split a clip with the keyboard?: Press S with the playhead where the cut should be. Backspace then deletes the selected clip or zoom.
Is there a shortcut to add a zoom?: Yes. Press Z in the editor to add a zoom at the playhead.