Is it somehow possible to rename/delete/copy files when saving a file and the Finder window pops up? I know you can do it with Windows, is it possible with Mac OS X?
If you are talking about operating on files in the folder structure you see in the Save window as if you were working in a Finder window, the answer is no.
The only operation that approaches this is to click on a filename appearing in the Save window, which will then rename the file portion of the filename, and save the file in its native format.
Below I'm saving a TextEdit file, (Untitled.rtf is the default file name & extension):
When I click on the Word document 3D_Course_v2.Doc, the filename changes to match, but keeps the format of the file, as evident by the .rtf:
A quick test in OS X 10.10 shows that you can copy and delete, but not rename files from the save dialog box. Right(control)-clicking on a file brings up the options for Move to Trash and Duplicate. The usual behavior of changing a file name by clicking on the already-selected file does not seem to work as it does in the Finder.
A commercial extension called Default Folder X provides this. Tested against OS X 10.10.3.
It has a few other nice features, like letting you pick an open Finder window as the save/open destination.