Can I split a 60 amp split-phase circuit to multiple 15 or 20 ampere circuits?
You should make a plug-in subpanel. Remember to keep ground and neutral isolated in the subpanel.

You don't need to worry about balancing loads - if the loads are balanced, the current in the neutral is zero, otherwise it's more than zero, in any case it's fine if the circuit is properly wired. This is the way it works into your house/building and at every sub-panel feed.

You want a sub-panel so that there are more-reasonable sized (15-20 amp) breakers on your 120V circuits, rather than a 60 amp - unless your 120V circuit is a 60 amp circuit with all conductors properly sized for that load.