You want ONE flooring. In the entry foyer, no matter tile or stone, or wood, you need a walk off mat. You'd not want rain, snow grunge tracked in or sitting there no matter the surface!! There are dogs on hardwood, whether engineered hardwood, or sand on site hardwood at double the lbs of your dog. Some have two or thee dogs. You can lay heat under engineered flooring, however, I see no reason. Wood is warm by nature. If that home is on a concrete slab, engineered hardwood is your is the choice- pre finished or unfinished.

Wood, no matter the variety, is a surface MEANT to be walked on. It also begs area rugs for style, for defining seating areas, for a good look in your home! You appear to have none?

From the pics, I might not suggest LVP ( vinyplank ) .....but its an option as well- can not be heated...but it does stand up to dogs and also can go everywhere. that said, nothing beats the real deal sand on site which might last 200 years.

ONE FLOOR on your living floor. : ) Bedrooms are personal choice, but nothing beats wood in combination with a generous area rug. It will outlast wall to wall carpet by ......................eons.

Keep it simple, use wood as mentioned with rugs. If, you have a formal mud room. Tile or stone can be used there, along with a radiant heat system. Up here in the northeast. Radiant heat isn't used much. Mostly just in bathrooms. HVAC's system usually heats the home. It also, provides good air filteration. So, you're getting clean fresh air compared to other systems.

I understood you wanted to do all one floor so do that everywhere . Good mats keep the entry clean and a 30 lb dog is not an issue just regular nail clipping, You can have in floor heating with hardwood or even laminate. I think consistant flooring is just a better look all the time . The stairs the same wood stained to match same for hand rail BTW you can use LVP with infloor heating but from what I see of your home the wood is probably the best choice .BTW I ahve hot water heat with rads and will never go back to forced air , no dusy vents no cold spots just better .

Agree on forced air........HORRIBLY uneven heat.

stick to one flooring .

One flooring, probably an engineered hardwood since you have underfloor heating. Paint the spindles and stair nosing white to match the trim, the handrail could be painted black or stained to blend with the floor.

Hydro air systems are a top choice in Eastern Mass. As long as your HVAC guy calculates proper vents and intakes. There are no cold spots to be expected.

Flooring decisions are not made in a vacuum. All the elements of the style of your home will affect your selection.

Decide how you want to address that major design element of the staircase BEFORE you make flooring decisions. Will it get a facelift or perhaps just a paint job?????