You need to check first with your HOA (if you have one) and also with your local building authorities. In general (and this can differ by municipality), you do not require neighbor's permission to build something on your own property. But you do need to check into the legalities of building such a structure.

Have you already broken ground, or are you still in planning stages? Hard to give specific advice without seeing your floorplan and elevation drawings, but here are some general thoughts in response to your questions: 1) if the foundation hasn't yet been poured, the builder can pour a wall across the outside edge of the patio (I'm assuming your patio will be in a U-shaped alcove like in the picture above). Then the void underneath is filled with 3/4" clean gravel, and the patio is poured and tied into the wall with rebar. It's also possible to drill into the foundation wall and "pin" the patio to the house with rebar. The issue is that as the backfill settles, you get the sort of big voids that critters love. 2) In many cases, your builder gets better pricing from subs because of the volume of business they do with them. That said, unless it's a ginormous area you're probably talking a price difference in the hundreds of dollars on concrete. If you're doing stone or pavers, the builder's guys are not always the best. 3) A year? We find that the area around the foundation of a new home continues to settle for 5-7 years. There are a couple of ways you can have the patio built where settling isn't an issue. You can excavate all the way to the house footer (undisturbed soil) and backfill with gravel all the way up to where you need to pour the patio, Or, the contractor can dig to the house footer and either pour concrete support piers with sonotubes, or they can build support walls. In either case, the patio slab is tied to those supports with rebar. 4) Depends on what the wall behind the stone will bear on. If you do a lick and stick veneer stone over concrete board screwed to a stud frame lagged into the deck posts (and a nailer on the house walls) you should be fine. But if you try to do something that bears on the ground, you'll have the same settling issues - unless, again, you dig all the way down to the house footer. Stuff like this is why I suggest hiring a landscape designer long before the house is built :) It's not that the builder is doing anything shady, they just try and stay within what they're comfortable with.

Your yard has a lot of potential. You might consider adding some planters to add some "nature" to the space. Would add a flat roof in the corner large enough to square the corner, extending it to the outer edge of the roof over the water heater, if practical. Painting that water heater the same color as your home is a good idea and you might also consider adding an "L" shaped lattice privacy "screen" (also painted the same color as your home) to hide the water heater ... sturdily built but created to not interfere with the working of the water heater and fairly easily moved out of the way should the water heater need maintenance.

Is anyone familiar with the porcelain tile by Emil called Cornerstone in the slate black? I love it and want to use it for our covered patio but concerned if it’s too smooth (i.e. as in slippery) bc we have a pool. The pool deck which is travertine is between the pool and covered patio so the porcelain tile won’t be right next to the pool. But, my contractor originally recommended a porcelain paver by Landmark which is nice in appearance but has the texture of concrete or asphalt….rough and chalky like which I don’t like especially for the covered patio. I much prefer the smoothness of the Cornerstone. It looks just like the natural slate (which I love but don’t want to deal w maintenance of a natural stone) but it’s porcelain. It’s a tile so 3/8” thick versus the 2cm thick pavers that are rough. I know it’s fine to use outdoors on a patio surface w a foundation which we have, but is it ok to use with a pool in the backyard? It says it’s “low slippage” but it’s not slip resistant, per se. Appreciate any feedback! TIA!