Is there any type of software in which you can animate the way the .stl object will be printed?
I'm not talking about what the end result looks like. I'm talking about a tool which acts like it's printing the given object as an animation.
I know it somehow depends on your printer but is there anything I can use?
You may wish to consider Craftware for your purposes. It's a free program in beta form that does provide a tool-path animation for printing the layers. It is not so much specific to a printer as it is configurable for your own requirements. The video shows what I believe you are seeking at about the two minute point.
Simplify3D also provides such information, but is not a free program.