I want to use ArcGIS to map my apartment. How can I create a projection file (with which parameters)?
You do not need a projection if your coordinates are arbitrary. If you are going to use a basemap to pick a corner of the building provided by Arcmap then you are stuck with WGS84. Autocad offers free trial and complete student edition if you are a student. There are many open source drafting programs to choose from but your question does not state an OS so I can't make a recommendation...either way that's an easy search and I am sure it has been covered here.
You could also draw it in paint or some other drawing program and geo-rectify the image in arcmap using satellite imagery.
Here is something to keep in mind: GIS is a poor architectural drafting program and AutoCAD is a poor GIS program.