Fujitsu, LG, Mitsubishi are supposed to be good brands. You can look around at websites like ecomfort.com or acwholesalers.com to get an idea of selection and pricing. There will be a huge difference in actual price if you have a contractor buy and install the system. For 5 zones, it will probably cost more for the mini-split setup you're asking about than to have ductwork run and a central heat pump system installed. A 2 zone mini-split costs about the same as 2 individual mini-splits, so you don't really save any money, just less outside units to make space for.
If you can describe the house layout, we might be able to give better advice. You might get away with two zones. One wall mounted unit in the central living area and another short-ducted unit that can placed up at the ceiling of a closet or laundry room that is sandwiched between other bedrooms.
I live in a similar area to you--- Blue Ridge mountains of Virginia. Our house is 800 sq ft and heated/cooled with one 1.5 ton Mitsubishi wall unit and it is fantastic. In another house (rental for now, soon to become primary home) last year, I installed a 2 ton LG wall unit that takes care of a 1,700 sq foot ranch, though the two bedrooms at the far end are about 10 degrees cooler in the winter and the doors need to stay open. This spring I'm going to be installing a 1 ton Fujitsu short ducted unit to serve those two rooms. For the Mitsubishi, a contractor purchased and installed for about 2x the price ($4,000 total cost, 5 years ago) of what I could have done myself. I did the LG myself and hired a technician to do the final connection and vacuum, saving me probably $3,000. I'll do the same again when hooking up the Fujitsu.
edited to add--- as Mr. Green said, Mr. Cool is an option, though I don't know much about their quality vs the brands I mentioned. I also have the idea that Mr Cool isn't as efficient in low temp operation as others. If you were to the DIY route, I'd recommend finding an hvac tech that's willing to work with you and get a professional brand. My guy charged $300 for his work. And make sure you find someone first, before deciding on going that route.
Also, the brand you end up with will likely be dependent on what the contractor is most familiar working with. Options like hyperheat (mitsubishi) aren't necessary for your climate.