Porcelin/Cermic will just need levelling and fit.
Kandean Rigid Core or any other Click or Glue LVT will need:
1 coat of Ardit NA or 1200 Screed Coat
1 coat of Ardit DP-C or F-Ball F77 Liquid Dpm
1 Final screed coat of Ardit NA or F-Ball 1200 or a screed of your choice with appropriate Primer...
For best floated results, 4mm Ply the subfloor and float the Engineered over it although judging by your photo, it looks like you could float over the floorboards as they are with a GOOD quality underlay like Duralay Multitex/Timbermate XL.
If you decide to ply, it might be worth going 6mm...
If you go down the Liquid DPM option, you absolutely will need a layer of screed over it as it is impossible to apply 'flat' So you will add some height.
If you can't let in some plasterboard etc, then I read an idea on another forum where you could rub the wood back and then use a Automotive Body Filler product, sand back and decorate over that?
If the tiles are properly tightly fitted and going nowhere, you can screed over them with something like F-Ball 1200 or Arditex Ardit NA self levelling compound. You need to consider that those tiles MAY have a high Asbestos content (most are fairly benign if they are Asbestos tiles but some...
The fitters will pin the carpet under the lip, and use a seperate piece to band the riser in two pieces.
They could use a single trim on the short length but it will look a little unsightly.
You can bulk the Ardex NA up with granite chippings then do a final coat over the top, provided there are no moisture issues of course (NA is moisture tolerant but it will depend what flooring you are laying over the top as to if it will stay on the floor if there is residual moisture present in...
You can't float a solid tongue and groove floor so you will be sticking it down. You need to 6mm or 9mm Ply the wood side of the area, then Liquid DPM the concrete and screed to level of the Ply. Then Glue the solid wood flooring direct to the subfloor.
Assuming you have not had anything on-top of the vinyl when these have appeared, it's most likely the dis-colouration is coming from underneath, often a water leak.