You could trace it again in Illustrator. The logo is very pixelating. Photoshop is not a tool for designing logo. :)

Yes, it is possible but will take some effort.

Here is a quick rework showing the 'white' areas as 100% Magenta. Doing this will give you the ability to output CMYK separations from Quark. (The M plate will produce the 'white' and the K plate will handle 'black')

It would be ideal if you could lay down the white base and overprint the entire logo on top, without any trapping.

If trapping is required in all of those areas you have the potential for a real nightmare because Quark can't trap an imported EPS or Photoshop generated bitmap.

Handling the trap in the EPS with Illustrator is pretty straightforward but working with that bitmap in Photoshop to get the trap right would be involved. You would also need to work in CMYK with two layers. Bottom layer 100% Magenta and the top layer 100% Black with the layer set to 'Multiple'. Always best to check your channels when doing something like this. (Hint: The Y and C channels should stay solid black if you did it right)

I can give you more details on the process if you can tell me how this will be printed... (eg. White + Black on a T-Shirt??)

Now why does that coloured version suddenly make me think of Prince Charles?

:->
Prince Charles meets the Juicy Salif.

Charles would spell separations correctly.

You guys are funny. :P

It look like parachute though. What is it kelt101 ?

You guys are funny.

No, really. If you want to get some respect as a professional, it helps to know how to spell the technical terms of your trade.