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.
Ok now with the photo that cleared things up, the fusable plug is a inexpensive overpresure protection. I haven't seen that in the past I was thinking you had an electrical plug that failed but since I hard wire I figured it was something I had not seen. Once you have the replacement installed. It would be a good idea to replace the filter dryer. pump the system with a little r22 to be used as a leak check gas then nitrogen up to 100 psi and check for leaks. If no leaks check for acid in the oil since the system has been open to atmosphere. No acid pump the system down below 500 microns. Failure to do this you will be trashing the compressor due to atmospheric moisture that will cause acid and that acid eats the varnish off the motor windings then the compressor is toast. Verify the system will hold at 500 microns or less. If the system holds a vacuum you are ready to recharge If it is not holding keep the vacuum on the system under vacuum as it may be the water boiling out of the oil. Once a stable vacuum. Recharge by weighing in the charge and it should be good to go.
If the acid test shows positive the system needs to be tested with a megger in fact this could be done at the very first step to verify the compressor is good. Low value from motor lead to ground the windings are already toast, I use 10 mega ohms to ground as a minimum anything over 50 mega ohms at 500v is good. Added to answer question to long for comment.I use quick shot acid test if the paper turns red it has acid you can use the pressure of the n2. If the windings test good to ground you can flush quick shot makes a product that works well, I would be berry surprised if there is not a filter dryer some are not much larger than a 10$ cigar , sorry that's the only comparison I can think of. With acid in the system I don't think I would recharge until the system is flushed as acid in a running system can eat the varnish in less than 72 hours and to recover you would need to recover with a filter dryer prior to your recovery machine to protect the recovery machine then flush the machine.
Best Answer
I would call the support line to verify the length. The unit is precharged with refrigerant and too short a line set will cause an overcharge condition. Overcharge can cause icing of the system and shorten the service life.
Worst case, make a loop close to the compressor. A loop helps reduce vibrations transmitted through the line set when the length is short.