Question: why are you asking? If you have previously shot film and want to know equivalents, fair enough. If you're new to all of this and moving straight to digital without having gone film, you don't need to think about it.
Firstly, the focal length is a lens characteristic and never changes with the sensor size. Only the effective field of view does. A 50mm DX lens is exactly the same focal length as a 50mm FX lens. As you correctly say, a 50mm lens on DX will give the field of view of a 75mm lens on FX.
The minimum focus distance is a lens characteristic, and wont change with different body.
Depth of field can get rather complicated, but as a rough approximation, people often apply the crop factor to the aperture value too e.g. a 50mm f/1.4 lens on DX would look similar to a 75mm f/2-ish lens on FX assuming the camera position is not changed relative to subject. Not exact, but it gives you the idea. Note that is only for effective depth of field, and does not apply to the exposure calculation, where the actual value still applies.
The way to think about this is, the lens doesn't know or care what is collecting the light at the other end. So if it is a lens characteristic, it wont change with sensor. This applies to things like minimum focus distance.
_________________Canon DSLRs: 7D2, 7D1, 5D2, 1D, 600D, 450D full spectrum, 300D IR mod
Lenses: EF 35/2, 85/1.8, 135/2+SF, 28-80 V, 70-300L, 100-400L, TS-E 24/3.5L, MP-E 65, EF-S 10-18, 15-85
3rd party: Zeiss 2/50 makro, Samyang 8mm fisheye, Sigma 50/1.4A, 150 macro, 120-300 f/2.8 OS
Compacts: Sony HX9V, Fuji X100.