Are the focal length, aperture, and sensor width attributes tied together? My question is, basically, if I were to use an aps-c camera (23.6mm width) with a 35mm camera (53.55mm equivalent) at f/4 (f/6.12 equivalent) do the settings go:

Sensor width: 23.6mm
Aperture: f/4
Focal Length:35mm

Or:

Sensor width: 23.6mm
Aperture: f/6.12
Focal Length:53.55mm

or some other combination I haven't thought of?
I presume you mean a 35mm lens on the first paragraph. Yes?

The lens would be the same lens as with the aps-c camera with the same aperture. if you use a sensor width 23.6mm:  aperture at f4, length 35mm. That would give you the same field of view and the same depth of field.