Looks like your existing island has a side outlet. That's a good option unless you want to do an angled power strip bar underneath your new countertop overhang. Aside from that, you're looking at a pop-up (Mockett's new low-profile round one is my favorite, similar to the Hubbell waterproof models), however that interrupts the flow of the countertop and is usually recommended as a last resort. There are a number of ways that you can incorporate receptacles into cabinetry or cabinetry fillers that have been notched, but that's on a different level of complication and takes up storage space. There are many threads here at Houzz with ideas for in-cabinet receptacle designs.

On the wall cabinetry, if you don't want receptacles cluttering your backsplash area, you have essentially two best options based on my own research for my own renovation: Legrand Adorne Under-Cabinet Lighting System. The second is power strips with lighting from Task Lighting by Hardware Resources. I'm sure there are others, but these two are the most prominent in the USA market for what you are wanting to do, raise all the receptacles, switches, lighting controls, and lighting up underneath the cabinetry. The Legrand system is more customizable for what you want to do, particularly if you want to control all or a series of your lighting from the same switch.

You can get short undercabinet lights. Even my kitchen which isn't so convoluted os yours I put an individual light bar under each cabinet. Your designer/electirican should have no problem with this. I don't see anything too complex about yours. The electrical will have to rough in the wiring at the proper location for each cabinet.

Make sure they rough in a box for the vent hood (get the manual out for the hood you want to use). Usually, they need a receptacle mounted up where the shroud around the exhaust duct goes.

As for your countertop receptacles, your "intent" is immaterial to the code. Every spot along the counter (measured along the wall) needs to be within 2 feet of a receptacle. The only exception are countertops shorter than 12 inches (the one on the extreme left MIGHT meet that, but I'd still put one there anyhow). There are a variety of decorative receptacles that can fit nicely in your backsplash. Another option is to mount them high (you can go 20" above the countertop right under the cabinets, the legrand adorne line might be interesting to you). The tricky ones may be under your double window if you need to put one there. I've got some turned sideways to fit under basements in my kitchen.

The island technically only needs one receptacle. Unless you want tombstones sticking up, you'll have to mount them below the countertop. Code allows the required one to be no more than 6" below the top and under no further than a 12" overhang. Mounting them to the ends of your design should not be hard and compliant. You might consider an additional one down low in the foot well on the "eating" side. My island in my former place had one down there, and while it doesn't meet the rule for the required outlet, it's allowed as an added one and was convenient at times.

@Ron Natalie I don't agree, my kitchen won't be convoluted, but thanks for the input.

I think by convoluted he meant, as you say, "the line of cabinets is so broken." I don't know about code where you are but I assume you can use pop-up outlets.

I have the Legrande Adorne system which I love. I also have a glass backsplash and didn't want to see outlets. What Ron Natalie said about the island is exactly what I did.