"1) what's the typical depth of the short-version apron front sink, so that I don't need a new sink base?"

The apron is around 6 1/2", the bowl is around 9". This is why the cabinet front (or stretcher rail on Euro cabinets) has to be removed. It won't let the bowl pass by. You can't put the sink in until the front is removed and you can't put the front on until the sink is in. They have to go back in together.

"2) I know you have recommended SS over cast iron and fireclay. I am actually a sucker for the white cast iron sink. To install that one, is Hercules sink harness strong enough or I need metal bracket under the sink to support?"

A HUSH or a couple Sink Straps will support a cast iron sink, but for ease of installation and adjustability, you can't beat a Pete's Sink Mount Kit. No wood blocking, I don't care what the sink manufacturer videos show.

"The sink just can not be replaced without removing the countertop. To save cost, I was prepared to just replace with another double bowl. But, this sink was sandwiched between the countertop and plywood (on both sides, about 3" wide), I can't pull the sink out."

The sink can be replaced without removing the countertop, I promise. Unscrew the cabinet front and the adjacent front(s) from each other and/or pull the dishwasher. Smack 'em off with a hammer and a wood block. Climb into the adjacent cabinet/dishwasher space, and with a multitool, cut about an inch or three off the sink base cabinet tops. Don't worry, you're replacing the missing cabinet with structural strapped sink. Now multitool the plywood off that's holding the sink in place. This really sucks, but it's doable.

After you've pulled the faucet and soap dispenser, drill a hole in a 2x4 that spans the cutout front to back, drop a bar clamp through the hole and through the drain hole in the sink, and loosely clamp to keep the sink from dropping uncontrollably. Drive a scraper between the stone bottom and the sink flange. When the sink drops, remove the clamp and pull the sink out the front. Recut, re-profile, and polish the new sink cutout. Install a HUSH or a couple Sink Straps to the cabinet sides. Slide the new sink in place, silicone, and re-clamp. Remove most of the silicone squeeze-out immediately. Put the fronts back on; you can't tighten the straps until you do. Hot melt glue the back of the fronts just like the factory did. Pull the clamp, clean up the residual silicone squeeze-out, reinstall the faucet, and you're done.

Unfortunately, without a tremendous amount of advanced-class work, you're not getting a Kohler Whitehaven short apron if that sink is in an island, which it appears to be. The "swoop" on the left side of your cutout is too deep. The Whitehaven maxes out at about 18 1/2" from cabinet front to the rear of the cutout. Your cutout alone is at 18". No way. I'll sell you a custom made stainless steel apron front sink with a faucet deck which I designed to fix this particular problem:

No swoop:

Whitehaven: