Any advice on setting up such a system, projects you can direct me to for inspiration - guidance. I would like to not go over 1000 dollars but don't let the price guideline deter you from making the best suggestion.
The main challenge would be setting up a weatherproof enclosure.
It's unlikely that you will be able to beat the weatherproofing of a GoPro, but you would have less ability to customize the automation if you go that route.
This setup runs in a small waterproof enclosure and runs a bit warm (35-45C) that dissipates any condensation or rain on the fisheye lens. This might help you with snow accumulation.
As I currently understand the question I would say that the solution mostly depends on what interfaces the cameras provide for remote operation, so without more information on the specific cameras you want to use your question isn't really answerable.
Are any of your old "digital cameras" dSLRs capable of running Magic Lantern? Is the 1000 USD (≈6500 DKK?) budget for the duration of the project, or just the set-up? Will you be able to visit it periodically once running? Are you trying to do a timelapse photo, CCV, or watch some specific event?
If they were connected to power and USB you could have a computer inside the house that tells the camera to take a picture at a certain time and then copies that picture to some online storage. My guess is that there are timelapse apps that can do that already.
As for the outdoors part there are lots of weatherproof enclosures available.