Flooring – Should I install hardwood flooring under the kitchen cabinets

Is cutting out the flooring where the cabinets will be not an option? You don't mention it, so maybe there's a reason, but that's the method that would get you the best attachment for your cabinets, while still allowing the flooring to float and move as it's designed. You could make these cuts with a circular saw set to just the right depth.

For the peninsula, I believe cutting out the flooring is your only option. If you try to attach it through the flooring, you're either going to 1) pull it too tight to the flooring so the flooring can't float, or 2) not fasten it tightly enough and it's going to creak and move every time someone leans on the counter.

Your vapor barrier needs to extend beyond the edges of the floor to prevent moisture from getting to the floor. Install in accordance to manufacturer's recommendations.

The expansion gap needed also depends on the manufacturer's recommendations - different materials expand at different degrees. The point of the gap is to give it room so it doesn't end up buckling. If you can't find a solid recommendation then use 1/2" for the maximum protection - and if the trim you install doesn't cover the gap, use quarter round.