Generally, the range hood should be 26" - 32" above the range. Some manufacturers have higher or lower recommended heights, but the majority fall within this spectrum.

The typical distance between the countertop and the cabinets is 18". It's hard to tell in this photo, but the cabinets here may be a little higher, which should accommodate the range hood minimum clearance. Also, keep in mind that as the range hood is suctioning in air, it will basically cool itself - so as long as you have a powerful enough hood, and you don't cook on all burners at the same time, and you always turn on the hood while cooking, you'll minimize the risk.

Heat damage may occur if you turn on the cooktop and NOT the range hood, in which case heat will accumulate and damage the electronics / plastic components of the hood.

Another option is to recess the hood deeper inside the cabinet, which will decrease the risk of heat damage as well as create a "capture box" from which fumes won't escape. However, since the control panel will be obscured, this requires either a separate control panel, or a wireless remote.

All of our "Insert-Liner" range hoods, which are designed to be built into cabinetry, include a wireless remote.

As noted, required height depends on the manufacturer and if externally vented.
Also, we have designed cab doors that open to each side with a higher exhaust hood behind the doors.

What is the floor? Beautiful kitchen!!

Existing oak floor that was bleached and stained with a custom mixed stain.