We got a Mitsubishi Kaiteki[1], which is a mini-split from what I can gather. We just call them "air to air heat pumps" here in Norway.
Anyway, the install itself was about $400, took one guy a few hours like you said. Another $200 for the electrician which had to hook up a dedicated circuit for it as per regulations, though he did it as part of a larger job so would have been $300 as a stand-alone deal.
Total it was $2200. This is considered a premium unit over here, we could have gotten something cheaper but alas looks matter.
Thing is supposedly guaranteed to provide COP of 2 down to -25C. We've had such low temps here a several times (in fact past week has been around -15 to -17C with a harsh 85% humidity, biting!) and we've never had to turn on the backup floor heating so far. Though you can definitely hear it working harder when it's that cold.
Yeah, you can see the two units here[1], also has dimensions for them under specifications.
The plumbing is covered by a plastic cover, both for protection and looks.
We have a small wooden "shed" for the outside unit[2], to protect it from getting clogged up with snow and also to make it less visible. Both the plastic cover and the "shed" is painted in the same color as our home, so blends in nicely.
There are a few zero pipe mini-splits. They hang across the window frame and use 1 inch of height with an inside bidirectional radiator and an external one, SCrewed down to the window jamb and connected to 120/240 volts internally. They resemble a window A/C, but the rads are below the window level and are wider and less deep. I am getting a few next year. This is one of many.
https://www.greenbuildingadvisor.com/article/startup-promise...
If you're system is hooked up to "floor heating", hydraulic stuff that runs under your floor it's still a one man job but can be a little bit harder to do.
The hardest part of installing AC is the ductwork. Swapping out the outside + inside heat exchanger unit is a 1-person 3-4-hour job.
Sorry if some HVAC company scammed you, but it's definitely not a 3 person job, unless we're talking some commercial building 5+ ton units.