based on the description of your kitchen, definitely semi gloss, particularly with polar white, it's quite a stark white, so semi gloss will soften it

It is probable that your hall will continue to be dark, and that the lights will need to be left on, even once you have painted the timber-work. Perhaps check out some options for improving solar access - Velux sky windows, solartubes or Illume fibre-optics. Given the cost you will be looking at to paint = sanding prep, 2 x coats of sealer, 2-3 x coats of paint + labour (if no diy)......would you consider a wallpaper?? leave the timber architrave and wallpaper over the panel with a neutral-white seagrass wallpaper ??

Ours were a light-mid beige colour and I didn't enjoy them! Actually the darker the colour the more they show the marks. We now have semi polished tiles in the areas that kids use. Still a slight sheen but way more practical. A good tile shop should be able to talk you through the different types of tiles.

Its going to stick out no matter what you do. Is there no budget to reface the cupboard doors and drawers? My Only Suggestion is to not do a White Gloss and instead use a Ravine laminate or a totally different colour and use the colour somewhere else in the kitchen like the off cuts for a frame for some artwork. Or on the kicks. If its going to stick out, and it will, then go all out. The next best option is go to a joinery company ask if you can take some off cuts out of the bin, make a temporary shelf and save up so you can do everything at the one time. If you redo the kitchen you will end up redoing that cabinet to get a consistent colour. Even with colour matching there is still slight differences because one is new and the other is old.