Could ducks and geese be happy with a creek instead of a pond?

Hi everyone.
I’ve never kept duck or geese, but I’ve always loved the animals and hoped to one day be able to keep them. I don’t have any experience with ducks or geese, only with chickens.
We’ve recently purchased a house with 1,5 acres of land and I was hoping to be able to keep some animals on there, just for personal consumption (eggs) and to use the animals to keep the terrain cleaned and fertilised.
Now our land is rather sloped. It’s actually a little valley in itself, with 2 waterfalls bordering it and a creek running through the middle of the domain.
I was wondering if 4 or 5 ducks and 2 geese could be happy living here. I’d want them to roam free (perhaps work with rotating pastures) alongside maybe 5 chickens. There is no pond, but there is the creek (it has parts with a rapid flow, and some parts that are slower). There would be one flat area where I could develop a pond, but I need that flat area to grow my vegetables. All the other areas are sloped pasture or forest. You can find a little map of the domain attached. Light green is ‘weedy’ pastureland that I want to sow cover crops on that serve the birds. The part nearest to the house would be used for growing veggies. The medium dark land is currently covered in brambles and low shrubs, but could be cleared in the future to make more space for the animals. The dark green part is terraced forest, but since chickens are originally forest roaming birds, I want to give them access to that terrain as well.

My main question is: would the duck/geese be happy with just the creek (and a few other corners in the garden where there is a small spring seeping out of the rocks), or would I need to develop a pond?
I don’t want my ducks to just manage there, I want them to be happy and thriving.

Also: can you keep duck/geese and chickens in the same coop, or would they each need their own little house?

They would be fine with a kiddo pool. So I don't see why they would have a problem with your creek.

With flowing water you might be able to grow fishes in a "pond'. It would be nice if you can provide your own honey, eggs, chicken and fish on-site.

S Bengi wrote:They would be fine with a kiddo pool.

Are they really though? I mean I know they are perfectly able to survive with just a kiddo pool, but I'd want to create the best natural habitat for the ducks/geese. What I'm basically saying is, I'd want to pamper the ducks. My main worry is that the ducks can't really swime/dive in the water because it's either too fast flowing or not deep enough. I don't know to what extend domesticated ducks have a need to swim though. Maybe all they want is a place to wash up? I'd love to create a pond but I'm seeing limited options for on my terrain to do so; though I am in no means expert in where you can or cannot create ponds, so maybe I'm not seeing all the possibilities.

With flowing water you might be able to grow fishes in a "pond'. It would be nice if you can provide your own honey, eggs, chicken and fish on-site.

I do agree wholeheartedly. I would love something like that, but given my steep terrain and limited flat areas I'm not sure where I could fit a pond (if the government would even allow me to do so. I'm not sure about the rules on this in Northern Italy).

edited to rephrase.

Here's a girl who kept 20 ducks, I believe, in the garage of a rental property (I'll provide a better quote on her text about ducks and streams if I find it).You might find the 'Azolla' novel.

You don’t necessarily need a completely flat place to make a pond. You don’t always need an earthmover either. Check out this thread about building beaver dams https://permies.com/t/97743/Building-beaver-dam or this one https://permies.com/t/140530