Ability to crop and rotate, automatically add “crop” or anything else to the end of the name of the saved cropped photos, have changeble hotkeys,
XNView MP is what I use and it is amazing, but to change hotkeys you need a separate software that works with the program, HotkeyP.
I use IrfanView (freeware): it crops, rotates, you can save the new file under another name (not done automatically).
And the SW delivered with the used camera (Canon SW).
Can it also save cropped images with unique names automatically?
I think it will be difficult to find a program that will automatically rename the cropped files as you like. But you can simply do this: 1) copy the images into a new folder, 2) crop them, 3) batch rename on the cropped files (there are bulk rename utilities you can install; they allow to rename multiple files at once, using flexible rules with wild cards, e.g. *_copy.jpg), 4) copy them back to your original folder.
I use Gwenview (for Linux), which is super fast. It is especially important for me that the program remembers the cropping area I defined for the previous image, which speeds up things a lot.
Edit: Maybe this is easier: 1) Crop all files in a folder you want, 2) in your file manger, sort per “modification date” to make it easy to select only the cropped ones, 3) select and rename them with your batch rename tool.
It looks like it’s the auto file naming that is the sticking point in the solutions above. I just checked Photoshop which I subscribe to. In googling the question I found that the latest Photoshop does appear to automatically append “Copy” to the new file name in Save As. So perhaps you could use that and then do a batch rename to change “Copy” to “Crop” (or just know that “Copy” = “Crop” if that would work for you).
In looking that up and checking my own system, I realised my version is out of date and on seeking to know why found that the latest Photoshop needs Windows 11 which my computer can’t do.
So whether this is a solution for you depends on whether you have a Windows 11 computer and whether you wish to purchase Photoshop. I pay the cheapest subscription version once a year and don’t find it too onerous - I use it a LOT.
Lightroom does that, (not sure about hotkeys). It will even let you mass rename files in your library. I have mine renamed to add date to the beginning. You can set it to to add custom text to export.
You can batch rename images very easily, with parameters that are specific to each image, if that is what you are asking.