The gap between the pile caps is not an issue. But it looks like the piles from one cap are too close to the piles from the other cap. Typically you need to maintain a center to center pile spacing of at least 2.5 times the pile diameter.
When looking at the stability of a rock slope the presence of any joints is an important consideration - particularly the strike and dip of the joints (ie. the direction of the joint).
It may not be as simple as you think.
I do not think anyone is going to be able to provide meaningful comment without first reading the geotechnical site investigation report and being provided more details (your sketch and descriptions are vague).
You can calculate the cracked bending stiffness using the method of transformed sections.
There is an example here: http://www.ce.memphis.edu/4135/PDF/Notes/Chap_5_2_flex-trans.pdf
That "load and support condition" table has nothing to do with K values.
If the top is restrained against rotation you can use K = 1.2. Otherwise it is K = 2.0.
For blockwork walls, which may be load bearing or non load bearing, the reason to show both on the same drawing is because the same sub-contractor will be supplying/installing them - so it helps with procurement.
Similar for precast walls.
I am a structural engineer so take my advice for what it is worth.
For normally consolidated clay the k0 value is calculated the same as for a granular soil as (1-sin(phi)).
k0 is the same for drained and undrained.
I had a similar situation recently with an existing concrete column needing to be removed.
The contractor cut through the column using diamond wire sawing technology and removed the column in several pieces. It worked well.
The method is similar to this...
You could look at active wedge theory.
https://www.researchgate.net/figure/Gravity-retaining-wall-with-narrow-backfill-a-Mechanism-1-and-b-forces-acting-on-the_fig1_346138861
Good point. If you take into account the vertical gravity load then it is probably worse than that.
I believe the Figure out of the commentary contradicts what is stated in in 14.7.1(b).