Is it really necessary to keep the paint bucket? And perhaps strong big shelves above would be better than counter top as can use top of washer for sorting - can you fix floor or remove door and replace with perhaps a blind above you can pull down when wish to hide sink?

Paint does go bad after a while, especially if it has a lot of air space (eg big container only partially full). I would start there and see if you can pour any usable paint into a smaller container, or ditch the whole thing if it has gone off. As to your other considerations, I would lean toward a space for clean clothing and ironing.

The paint isn't mine or my partner's. It was left when his mum and her partner moved out. I've got no idea how old it is, but it's darn heavy!

take everything out of the cupboard then clean the whole space and get rid of anything that you don't use (MIL may have some space to store the paint in her new place or better still, if the paint is still useable, use it! (she may have tax deductable allowances for paying you to do some maintenance in a rental property) and then get rid of anything you don't need and, after working out the sizes of everything that you have to keep, rearrange the existing space to suit what you need to store and/or add an extra shelf and some wall hooks to make a huge difference until you decide to do a proper renovation on the whole space

1. Agree with oklouise - remove everything, clean, ditch if not used in last 12 months
2. Identify what are you top three priorities for this space eg a) washing machine b) dirty laundry c) drying clothes/drip dry
3. Then consider options for layout.
A) would suggest washing machine left hand side (people work left to right, top to bottom), utility sink right hand side. Counter top across machine. Cupboards installed across width and to the ceiling. Installed rail under cupboards for drip dry or drying off and then add down lights in base of cupboard (if possible so can see if stains have gone, and feel like not in dungeon spacing away)
B) add storage bracket to wall for ironing board to hang with iron. Washing machine left hand side. Run cupboards from top of washing machine to ceiling. Bottom part of cupboard have two baskets for whites and colours (option to have door off of course). Rest of cupboard for linen storage. Sink right hand side. Cleaners/chemicals go under sink. Open shelves with baskets slotted in from head height up to ceiling on left hand side.
Sure you can think of more

That's far too much of an investment to be called a facelift

All I'm doing is fixing the shelving and maybe installing a countertop with an open question as to whether to leave the washing machine in its current spot (under the counter to have folding space) or lift the machine chest height and have a shortish counter underneath with sorting bin. The scope is deliberately minimal.

The paint is not the only thing left behind. We wanted to BnB one of the bedrooms after Lee moved out as my partner has an ongoing illness and we're living on my income. There were boxes piled atop the bed, all left behind, too many to fit in a closet, and we've been told we just have to deal with them.

There's a lot more to this story - wall oven that hasn't been working for months, a rotting kitchen bench with weekly promises of a new one next week. In short, if I knew we were staying there long enough to mentally benefit from the investment I would consider it. Given my reality, though, anything as significant as hiring contractors to flip the plumbing with the powerpoint is out of the question. It's about self-preservation.

disappointing that your landlord is not more responsible and more difficult if you're at work and your partner is not well: if there's no legal agreement about who owns what and you've been told to "deal with it" disposing of anything that you can't use or is not obviously valuable...it's not safe to raise the washing machine without a specialised washing machine stand and it will be more comfortable left on the floor to use the top of the machine as work space... try the simplest cheapest options and keep only the essentials on a shallow lower shelf and hopefully the top shelf may have room for some of the culled boxes from the bedroom...check out the various shelf options in the on line Bunnings etc catalogues ..unfortunately the paint can't go in the bin so make sure the lid is secure and with the help of some hefty friends carefully enclose the paint can in plastic garbage bags (in case of spills), load it in the boot of the car (supported by something to keep it upright) and take it to the nearest recycling garbage tip...hopefully the worst that will happen is the owner will need it replaced if/when she wants to do more painting.. if you need an oven it could be worth finding out the cost of a repair but it could also be no more costly to consider buying your own convection microwave and use the old cleaned out unworking oven for storage of some of the stuff out of those boxes...for the rotting kitchen bench there's a product called "builders bog" that can fill in all sorts of horrible problems and then (carefully covering all the other surfaces use a cheap can of (red?) spray paint to at least make the bench look a bit better and, one box at a time you can take control of the situation, best of luck to you

Your money should be spent on the overhead storage. I would look st ikea for options maybe a shelf made of bars you can then hang things from below and put baskets on Given the trough on left and washing machine on right a bench is basically provided by the washing machine. It would be designed to allow a dryer on top so it can take some weight. Personally given the other issues you have I wouldn’t spend money on bench.

if you are paying near market value rent then you deserve s better deal. Force the issue with your partner and get his mother to remove her crap or bin it. She should also fix the floor and oven. or move out ! Good luck