IMO get the range out of there completely and as for the rest I have no vlue since you will need to know what is hiding in the 1/2 wall if aanything at all.

There really isn't any room else where. you could switch the dishwasher/sink and range, but that is now reworking plumbing and adding a lot more cost. True we don't know what is hidden in there, but was hoping folks might have perspective from other oven ranges where there isn't a full wall behind it like on an island.

Is the range approved for a gas connection at its base? What scheme do you have in mind to meet the requirements of electrical outlets supporting island tops if you remove them from the half wall? Does your plan include cutting off the countertop overhang that the half wall presently supports? What does your range installation manual require re combustibles at the back of the range. I suspect you can only gain about 2 inches.

Some here will want a dimensioned plan view to consider.

And how does cooktop effluent get out of the house?

What you need behind a range in an island is space -- otherwise it could be dangerous if people are there while you're cooking. I had a range in my kitchen island and I loved it (miss it terribly), but I had about a foot of counter space behind it, and no seating at the island. You also need an island exhaust fan, since it seems you're not close to a window.

Or if you're planning downdraft ventilation, you'll need space for the ductwork.

My brother did an island with no half wall, but it was cleverly designed and holds the sink, not the oven.

@Toronto Veterinarian @kaseki Thanks! I'm in the very early stages of planning, so I don't have a range picked out yet, but the current range has down draft that doesn't vent outside( if I had to guess), I think it just recirculates the air. Literally no brand identifying on current range so can't just google current specs. Adding an exhaust hood above, or duct work below could be $$$ so would like to have new one have downdraft with recirculation. There is already a gas connection for the current range, no clue if it's approved for that or what electric scheme I may need or what it looks like back there. This is what I'm trying to figure out, is there any solution where the wall isn't necessary or would gas/electric make that not possible? I plan on extending the new counter 12 inches for true bar seating so space behind the stovetop isn't an issue to me.

Generally, one wants the bar's 'customers' to not get grease splashed, or even boiling water splashed. This requires distance and usually a change in height. (There is probably a trade-off there, but I don't know what spatter angles and distances are to guess at it.

You need to meet code requirements for electrical outlets. (Get a copy of the NFPA 70 Handbook.)

The over hang needs support, usually, and this support has to transfer the load to something -- maybe the cabinetry, or else a half wall with angle iron over it. My bar part of a peninsula is supported over a 6-inch half wall six inches higher than the cooktop counter, and uses 2 x 1/2 angle under the granite and extending down the support wall on the cooktop side.

Stone, if used, should be supported to resist human weight on it at any point.

Hot cooking requires decent ventilation, and I think @opaone can regale you with what a gas cooktop needs to achieve air quality.

If your gas connection is in the half wall, it would have to be moved and the range approved to connect to a floor fitting.

If electrical were to feed the peninsula from the attached wall via a small "soffit" under the bar top at the back, one might be able to avoid the half wall thickness needed by the electrical. You still need to meet having electrical accessible to every conceivable placement of electrical stuff on the cooktop or bar top.

In general, I can get away with bar seating due to height and distance, and to using only induction cooking under a very effective hood. You may be better off to reconceive the living room/dining room configuration.

Downdraft ventilation will not adequately capture and contain cooking effluent. Even without splatter the bar will be unlivable if serious cooking is in progress.

Do you have a measured floorplan? It looks like your stairs come up where you're planning this "addition". If you add another, say, 15" to the depth of the counter, will that block entry to your living room?

What's the current distance of the countertop from the range to the edge? It looks less than 12" which is not ideal for that type of set up. I'd want at least 15" between the range and the bar edge.

Your range appears to be a Jenn-air/Maytag, should have tag perhaps behind the lower drawer or or frame. If you don't see a filter in lower drawer area maybe it is exterior vented. The current Jenn-air website shows a downdraft dual fuel, gas top (electric oven) downdraft range available with an optional vent free filter kit but if your current range is all gas that complicates your renovation having to run a 240 outlet if there is not one already.

That arrangement doesn't make any sense at all. The oven/stove needs to be on the back wall, where it can be vented to the outside, and your sink moved to the front peninsula so you can have a real sized sink instead of a bar sink. Or move the peninsula to the side wall so you have an L shaped kitchen.