Sizing circuit for mini split heat pump

Use a 20A breaker and 12 AWG (or larger) wiring.

I read through the installation manual, which was surprisingly bereft of meaningful electrical information. Regardless, the MCA is the required Minimum Circuit Ampacity, meaning your circuit needs to be sized appropriately for at least that amperage. The 24kBTU unit has an MCA of 15, so you'd need a circuit capable of supplying at least 15A.

Because this is HVAC equipment, a 14 AWG circuit and 15A breaker is insufficiently sized. This is a continuous load, so you're only allowed to size to 80% of the rated ampacity, and the next larger size is 12 AWG and 20A.The Max Fuse simply states that the maximum size of the OCPD should not be larger than 25A.

HACR is likely to be listed on your breaker, but it's no longer required. Neither the NEC nor UL requires it to be labeled anymore, so don't stress over that.

You can do what others have suggested and use a larger ampacity circuit with an appropriately fused disconnect at the unit, but your circuit would be needlessly oversized.

To directly address your questions, the labels indicate electrical specifications for the unit, and if the instructions only show the indoor unit directly connected to the outdoor unit, then the outdoor unit should provide for any protection the indoor unit requires. The circuit only needs to supply enough current for the unit and not allow more current than the maximum rating.

And no, you don't need to use fuses over breakers.