design standpoint the extension structure is just leaning on the existing structure to allow settlement. then it is a matter of just filling up the cracks. it does not join up with the existing structure as a whole as differential settlement might complicate matters. This kind of extension we usually recommend bakau piles ONLY if the water table is high as it is prone to rot if not covered by water. Your SI should indicate the water table. RC pile is not practical as the shockwave it generates will cause your neighbours problems. The A frame rig you assumed to be for piling is only good for bakau, not RC as the hammer is not heavy enough to drive the piles to set. The pic you used is NOT a piling rig. Even if used as friction, doubt it will go more than 3m and that might not generate enough friction for load bearing. This kind of reno is bread and butter job for C&S consultant so find one. Design fee used to be under 1k depending on locale, plus misc fees around 3-4k. Once you have your construction plan, you can find a contractor to do the work. My guess is it would be bakau piling with pad footing. Just in case the piles fails, the footing is still able to provide support. Cracks around the new and old building interface is normal.

i made a mistake. looking back at your drawing of level, noticed it is backfilled and using reduced level. Assume needs to go 3-5m below RL0 to get the sufficient friction. If your backfilled height is RL 8, then add another 5m means need a depth of 13m. Normal A frame rig uses less than 1 ton hammer while a proper one uses at least 2.5t. 1 ton hammer prob will not drive to that depth and can be misunderstood to be set. I think the settlement in your area is due to the backfill not compacted thoroughly. Your developer should have at least 3 borehole logs done for the SI and you are most probably using the closest.