How to repair deck sloping on one side of the house

Lag screws should be fine. If possible, have them go into the ends of the floor joists as well. You can probably see where they are from the outside by looking at where the nails are in the rim joist.

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!