I live in Oregon USA 43° lat, live in the city and have ducks and chickens
Here are my responses to your questions:
1) Is it better to have a duck or chicken house with a weldmesh floor or solid floor (I live in the UK so we get cold spells sometimes)- I've heard they can catch their feet on chickenwire but would weldmesh be okay?
Do not use chicken wire, as you say it rusts out to easily and then is a problem for you and your animals. If you have predator issues do use welded-wire (weldmesh). I like to build up the area where the house is to go with wood-chips (creating high ground hill keeps the area dryer), next lay down the wire extending around the house a foot or two outside the walls (stops chickens from digging out their own house

. Then place/build the house on top and fill inside with leaves (not chips) about 1-2 feet deep - totally stops any smell, and chickens
compost this material up for you and your garden adding their own nutrients. In addition it is good to have the roof extend out 1-2 feet past the walls to further protect against the wet and weather. Oregon can be a wet place....
2) Can ducks & chickens peacefully coexist in the same run, or even the same house at night? Can they be fed the same feed or do they steal each others?
Yes, and Yes, and Yes..... You will have to address separate water and feeding provisions as mentioned by Cinebar above, but what you feed can be the same mix. I would add that when you feed ducks completely they are not the foragers (bug eaters) they would otherwise be. For this reason, pest foraging/elimination, I would NOT pen my ducks.
I start my ducks and chickens off together (heat lamp, food and all), but then at 3 or 4 months the chickens get introduced to the chicken house/pen and the ducklings are given free range of the backyard. I do not free-feed the ducks, just a cup or so per Indian-runner per day. This keeps them serious about cleaning my yard for juicy bugs. The feed increases a bit in the winter. They have their own crock for water and small plastic pool for swimming. I keep the pool on my patch of lawn in the summer time, dumping on the grass every day and refilling. In the winter I move them into the ivy, dump about once a week as the yard has lots of fresh puddles occurring naturally.
3) How often should their run be moved around to prevent parasite build up?
I do not move my chick house/run as my land/space is limited. However I add fresh materials to their area all spring, summer and fall - by the foot in depth (grass, hay, leaves, etc.). I hose down their house inside once in the spring and once in the fall (for dust mostly). I use nothing but water, organic materials inside and out, and a little DE in each next box when I change the shavings/straw. I have never had a parasite problem.
I also give my birds DE in their feed, and their feed is a natural mix not processed feed-store mix. Keeping the ground deep in litter naturally helps the birds to avoid their own droppings, and to recycle *grin* and compost. In the summer I add water to the litter in their run keeping the microbes on the job breaking down harmful bacteria before it builds up. And in the spring I remove the now dirt from inside their house and out using it for my raised garden beds, and placing new leaves, grass and hay in it's place. Anytime I need soil I go to the chicken area.
4) How much water would 2 ducks need (depth & width) for it not to go murky, and how often should I take them off & give them a paddling pool to prevent parasite build up?
Ducks continuously wash their bills in water. So any water will grow dirty/murky. I have no experience with ponds made for ducks so I cannot say what size you would need, or how often it's water would need to circulate.
Regarding water dishes and pools - I make water dish & pool flushing easy for me #1, and contribute to second use #2. This way changing the water is a win win. For example, in the summer I have to water a small patch of lawn anyway so I keep the duck water dish and pool on the lawn, I blast their droppings into liquid fertilizer, and blast out their dishes adding new water to everything as I go. This hands-on approach keeps me close to what is going on with my animals, yard and garden. By having a water dish that the ducks cannot swim in I insure they have much cleaner water for their needs than just a wading pool.
I change/clean out the water dishes in the winter for ducks and chickens daily or every other day, depending on the rains at the time. This is more than what is needed (over-kill) I know, but I like fresh clean water for my animals (just a thing with me).
5) What annual and perennial crops (I am East Coast UK) can I plant in their run for them to forage naturally, without them being totally decimated?
My chickens have taken out long established English Ivy, small shrubs and trees - so this I cannot advise you on. Unless you move your chickens regularly they will clear an area - from my experience. So chicken's I pen, ducks I allow to free range.
Ducks do need protection from larger predators, a dog or good fence works well for this, and they do not need shelter. Mine can be found during a harsh storm to be out in the middle of it resting and not seeking any shelter (crazy).
With my ducks and chickens my pest problems are gone and I have hard working composters so I am saved two hard jobs....on the urban homestead!
I hope this info helps get you started.