Can I swap between core and isolation and strength skills?

No, swapping between them will make you progress slower due to infrequent training.

Can I do strength skills at the start of the workout instead?

Yes, if you are especially interested in learning calisthenics strength skills and already have a solid base (at least 10 pull ups & 12 ring/deficit push ups). Then you can do strength skill work at the start of the workout instead, where you are most fresh!