I interpret the question to be not about tread depth or riser height, but the remaining, load-bearing portion of the stringer (dimension Z), and the height and depth of the upper stringer heel (dimensions X and Y). Here are some general thoughts...
The remaining thickness (perpendicular to the bottom edge) generally shouldn't be much less than it is below ...
Is there a minimum distance that a tread run needs to be on a stringer?
No. Nobody cares if the tread is on a "stringer" or not as long as the tread is dimensioned properly and supported adequately. An existing board that is already there and is adequately supported by a structure other than your stringer is perfectly acceptable, as would be an ...
OSHA 1910.25(c)(3) requires standard stairs to have a minimum tread depth of 9.5 inches (24 cm).
IBC 1011.5.2 requires that stair treads be 11 inches (279mm) minimum measured horizontally between the vertical planes of the foremost projection of adjacent treads and at a right angle to the treads leading edge.
IRC See Fredric Shope's answer.
OSHA and IBC ...
IRC says 10" minimum but there are likely exceptions.
I've heard that more important than meeting code-mandated minimums is consistency. Even with narrow stairs and short runs, the rise must be consistent from step-to-step (luckily, this is easy to calculate) because tripping is extremely likely with inconsistent rise over a span of stairs.
See the ...
Lose the middle beam
If the joists need support in the middle use bricks or masonry units or something else that won't rot if it's partially buried.
if you need to prevent the joists from twisting use blocking between them.
Given that you don't have joist hangers, the solution seems clear to me.
Set your three posts under three beams.
Lay your joists across the beams.
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Go to the library and check out a book on building decks. There are commonly accepted specifications and construction methods for decks, and a good book will walk you through the design and building processes.
Also, local building codes may have something to say about the construction of a deck.
How about adding an option (5).
The thinking is to evenly distribute the floor load to the edge beams, so the edge beams will be stressed less and deflect less than a concentrated load from the center joist. Since it is now stiffer and the space between joists is much smaller for the deck board to bridge over, the board can be thinner, and directly nailed ...
The 2015 IRC changes section R312.1.2 Height of guards. From 2015 forward the section on guard height "shall not be less than 36" in height as measured vertically above the adjacent walking surface or the line connecting the nosings." The 2015 IRC effectively rescinds the requirements to measure the guard height from an attached seating bench.
Actual X-bracing, not just a few diagonals.
If the ground is amenable, sink 4 posts to reasonable depth and bolt those legs to them. If not, more outriggers or guy wires. Or do both.
Beware the brace to what appears to be a dead rotting tree. Those fall over with some regularity. That might take you along for the ride. i.e. remove that brace, brace to ...
I've noticed the same thing. Measure the height at either end and cut a 2x4 into a wedge and move it toward the middle of the railing. Maybe you'll be lucky and it will be 90° and easy to raise the sag.