I cut a part out of one feature and added it to another feature in a polygon shapefile. My problem now is that both features has the same values in the area-, perim-, p2a-, width- and length-columns? The features are very much different in size, how do I get QGIS to sort of "re-calculate" the areas etc?
Use the field calculator (Layer>Open attribute table>Field Calculator/Ctrl+I or right click shapefile>Open attribute table>Field Calculator/Ctrl+I). There is an operator "area" that will calculate the area of each row in the table. All units will be calculated in the units of the projection, so you probably want to project it to a projection that uses feet or metres before doing that, rather than lat/lon.
The fields in the attribute table are only static values, even if you created them by using the geometry section of the field calculator. There is no way to re-calculate them automatically if you altered the polygon size, or changed the CRS of the polygon layer.
The only way I see is that you have to delete those "Geometry" fields, and re-create them by using the functions in the geometry section of the field calculator.