I created a new project from remote sources and entered my servers data. PhpStorm instantly began to download the whole Magento project, even though I only need specific files for development, e.g. I don't need all the images or cache folders for my purpose. Now it takes 4-5 hours to download the whole project.
In NetBeans you can choose which folders you want to download to your machine, is this also possible in PhpStorm?
Yes it's possible.
You should have marked such unwanted folders as "Excluded from Download" --
See official help page for appropriate wizard step.
Other ways of creating a project:
Just create empty local project and then configure the rest manually (deployment etc); once done use "Browse Remote Host" and download folders/files you need.
Another way -- download all needed files locally first using you preferred program (e.g. FileZilla) and then just point to the project root folder in "Open" dialog -- IDE will create new project from those files.