I have installed 3/8" floating laminate in the living room, and in the kitchen we have tile. The laminate is 9/16" lower than the kitchen tile. I'm going to make a threshold similar to this:
Is there a rule of thumb of how deep I should make this (dimension A) for a 9/16" rise, and dimension B, of how much over the tile (dim B) it should be?
I will be making this out of solid oak.
There really isn't. I'd consider an appropriate slope to be somewhere between 22½° and 45° from horizontal. Take into consideration the style of nearby moldings.
Practically speaking, the total height of the thing above the lower floor surface is what matters. It's a trip hazard regardless of slope, and therefore height should be minimized. Just keep in mind that the wider you make it the more visually conspicuous it is, and the more fragile it is. At some point it becomes a hassle to step over.
Regarding lap, it just needs to be enough to support the strip so it doesn't crack when the piano rolls over it, and covers completely even if the flooring shifts a little seasonally.
I just recently did a transition very similar to this. Instead of lapping the tile, I reduced the transition height by making the strip flush on top (level with the tile). I then used a color-matched silicone caulk to create a durable, flexible grout joint between the tile and the wood. It came out beautiful and there's virtually no trip hazard.
_____________________ __ ________________
tile |__| ______\___________________
^ |__________|| laminate
grout joint transition
strip
I'd consider doing that instead of adding height with a lap. It does require that you have a clean tile edge. I actually recut my tile edge with an angle grinder first so it was clean and straight. I also masked both sides of the grout joint before applying caulk so I could tool it into a cove without squooging caulk onto the strip and flooring.
