Can I add outlets using #14 wire on a circuit with #12 wire and a 60 amp breaker?
Breakers have to protect the wiring. Normally that means:


14 AWG wire must be on 15A breakers (or smaller, but that's the normal size in the US)
12 AWG wire must be on 20A breakers (or smaller)


If you put 14 AWG or 12 AWG wire on a 60A breaker then you run the risk of an overload of the wire - e.g., 40A - having no chance of tripping the breaker before the wires overheat and your house burns down.

The minimal fix is to replace the 60A breaker with a 15A breaker. But that will severely limit your power usage. The better fix is to split the circuits onto multiple 15A (for circuits that include at least some 14 AWG wire) or 20A (for circuits that only have 12 AWG wire) breakers. If you have space in your panel then that's easy. But if you don't have space - which may be why everything was combined onto one 60A breaker in the first place, then the long-term solution is likely a new main panel. But the short-term solution may be to use the 60A breaker to feed a new subpanel. The subpanel can then split up the circuits among several 15A and/or 20A breakers. Depending on the condition of the main panel, it may make sense to install a panel that is large enough to become a new main panel. You can then move additional circuits from the main panel to the subpanel until everything is transferred and then move the service feed to the new panel, toss the original panel and bond neutral to ground in the new main panel (as a subpanel you would not bond neutral to ground).