Can this be done? printing with 2 colors with a single extruder?

While this is low on the coolness factor, you could just glue the 3D printed logo to the 3D printed iPhone case.  You might even create in indentation in the iphone case to take the logo so it doesn't look glued on.

Printing in two colors could also be done by changing the filament while printing.  However, this would be hard in this case since the back of the iphone case is usually flat on the bed. You really need the logo on top.

A third option is to create a void (cutout) of the logo in the iphone case. A blue case with a white iPhone would show the logo in white.

Well, sounds like you want to print a separate file of just the logo and just the first layer. Then print a second file that has the logo missing and print the rest of the case. For starters, your second file would need the indent of the logo at the first layer height depth. The second thing to be careful about is that Repetier Host auto-centers STLs. So you would have to make sure that the center of your logo-only file is exactly the center of your case file. You also do not want the logo to be any higher than one layer as the nozzle would bump into it when printing the case.

As for if they will adhere, I dont see why not. Layer adhesion has more to do with the heat of the material being layed down. As long as you have a good extruder temperature, it should work.

Google "richrap color mixer" to find  multi color single extruder (multi motor) device.

The "pause and switch colors" trick is actually very easy, just a little time-consuming, but the results can be quite impressive.

All you do is figure out which layer should be the last one in the first color, and then find where the layers change in your gcode and insert @pause.  When printing pauses, quickly move the head using manual controls up and over.  Retract, change filament, run through enough to get to the new color (for those signs I didn't even bother, because black displaces any other color very quickly, and a little bit of "dark green" wouldn't be a problem anyway), then resume printing.  Repetier-Host automatically returns to the location it was at and resumes printing.  Easy-peasy.  I printed ten signs that way and they look really good.

But that only works if you split the colors by layers.  Splitting the colors within a layer, that's much, much harder, unless you get a dual extruder machine.

I you are good at editing G code... You could create three files, exactly centered the same. One with the first layer of the logo, the second with the first layer of the rest, and one with the complete item in a single color. Generate the 3 G code files, and replace the first layer in the complete item with the logo layer, a pause, and the rest of the first layer without the logo.

This is a very manual build of the G code, but should work. Probably difficult and time consuming the first time you do it, but who knows. You might become a G code guru.

Would be great if there was a tool to simplify something like this. If the slicer auto commented things like the start and end of each layer, etc, editing and debugging G code would be much easier.

Any reason you can't treat the logo like an inlay?  I've done this with my CNC machine cutting wood, and it works really well.   Imagine it work the same here.

Very true;  Just  wondering if you could do something like this here.   
That is have 2 files.  One of the main "background" with the object in the different color cut out. 
Then the other file the object that will be printed in the other color.
Then join the 2 prints.
Actually once I get my 3D printer up and running, I hope to do something very similar.   May be a while, but I'll post the results.