1X stock is fine if you have 16 inch on center joists, and the joists are themselves sized properly. Using 2X stock over undersized joists does help make the deck feel more solid. If you have 24 OC joists, I'd definitely stay with the 2X decking. An other option if you have wider joist spacing is to add a joist between each one if possible, then you can use 1X wood or even better, composite decking that will last a very long time, low maintenance and you can have your choice of colors and not have to worry about staining etc.
First off let me quote the American Wood Council PRESCRIPTIVE RESIDENTIAL WOOD DECK CONSTRUCTION GUIDE (which I recommend looking through).
MANUFACTURED WOOD I-JOIST: Many new homes constructed with wood I-joists include 1" or thicker engineered wood products (EWP) – such
as oriented strand board (OSB) or structural composite lumber (SCL)
including laminated veneer lumber (LVL) – as band joists (or rim
boards) that can support the attachment of a deck.
However, some older homes might be constructed with band boards that
are too thin (less than 1") to support a deck. In such cases, a
free-standing deck or a full plan submission is required.
You asked:
Question 1: I assume that if these clips are still being produced that it must be code assuming proper lag screws and spacing have secured the ledger board?
Yes.
It doesn't matter how the joists have bearing, whether it is a hanger, ledger, or beam. Just use the appropriate hardware/fasteners for the application.

Question 2: If I were to cut away the masonry and lag into the 2x6
studs for the ledger support, would this be a sturdy code compliant
ledger board?
Back in the "good old days" when homes were balloon-framed they would use let-in ledgers or ribbon boards to support the floor framing (which was also nailed to the studs, though). So, it is doable, but you may need special permission or even engineer's drawings/specs for approval (might also include blocking and extra hardware). Either way, I would say it is not exactly ideal, but at least you could use structural screws like Ledgerloks or GRKs, since they are smaller diameter and self-drilling and not nearly as likely to split the studs.

(balloon-framed, ribbon-board (ledger) and fire-blocking)

And finally, if applicable, if I were to cut away the masonry to mount
a ledger board, I would be worried about water possibly penetrating
and getting behind the decorative masonry.
You would have the same worries doing the same thing when cutting out siding for a ledger. Flashing to the rescue :)
Question 3: What is the most appropriate way to prevent water from
getting behind my ledger board or the masonry?
Again, flashing is the most important part (aside from proper fastening). You have to lap the flashing properly with the building paper. Just remember the direction water flows so the surfaces have to lap the same way a shingle would. See image below for a good illustration of proper flashing. You can also read this short article describing some of the flashing material choices.

After all that, I would say your best bet is to verify your rim board thickness and do a normal ledger. The second best option is a free-standing deck.
Just be careful there is a lot of bad advice and information on the internet!
Best Answer
That's a "breadboard end", and it's simply a design element.
There is no requirement for it that I'm aware of. If you do an internet search for deck images, you'll see a wide variety of decks with and without this type of breadboard edging, and the ones with it will have it in varying widths. It's purely aesthetic.
The most important thing is to have metal flashing that runs up under the last piece of siding above the deck and across the top of that tripled ledger board. The metal will help shunt water off the top of the ledger instead of letting it sit there and rot it out.