Really hard to comment on this without photos. Any chance you can upload some pix of the spaces in question?

Call an interior designer and a contractor to the SITE. Seems you deserve a better lit and more inviting laundry than what you now have. A closet would be handy as well. Why diddle with ten thousand answers, no dimensions posted and an unknown budget? Get on site help. A certainty of the games and piles and winter clothing storage amount is best determined before that on site visit. ......if not needed? Out.

Yep best way to go.

In the meantime, I'll join in on the mental exercises of thinking things through!

First question: Who will use that kitchenette/bar, and for what?

Will it be a place that people get snacks and drinks while they're hanging out in the basement room?
I would want that kitchenette to feel part of the enjoyable living space--so, I wouldn't necessarily want it shut away in the laundry room.

So I'd put it in the corner of the room that people will spend time in.

(but I'll also say--I would probably decline that fridge--it takes up a lot of space, and I personally wouldn't find it of enough value. If I really was in love with the idea of a kitchenette, I'd get a little bar fridge.)

It is not very clear from photos and floor plan.
If you want to have clean place and store all stuff at the same place - build sliding doors floor till ceiling and put some organizers inside.
It pretty easy and no mess, but pricy. Example on photos - we built mirror doors in this repurposed garage.

I’m not sure about kitchenette, better to invest in nice laundry: organize storage for clothes, storage for detergents and cleaners, maybe a kind of pantry.