Regarding color temperature of the projector, you need to know what kind of bulb it uses. I'm not familiar with them really, but I would guess that it'll be overtly blue. TV screens, as I've found out, are quite blue to film, when it looks grey to the eye. However, with a flat-screen computer monitor, which uses fluorescent back-lighting, it comes out rather green.
Now you say "If i discover that the colour temperature is something like 4000k can I simply project a pale blue image to achieve correct white balance?". I think the answer is maybe. By changing the image that you are projecting, you're really just changing the proportion of each red, green and blue "pixel". You are not fundamentally changing the color of the illuminant. However, in theory I think it should work, assuming it's a continuous light source.
Find out what kind of bulb it uses.