The problem with the Wii is that they sell fewer games for it per console than they do for the other two. The large proportion of MS and Sony's profits come from the games themselves.
That's not true anymore, actually. I think the PS3 has the lowest attach rate (in the US at least).
Sony and Nintendo were working together before the PS1 was released but they had a disagreement as Sony wanted to use discs and Nintendo wanted to use a cartridge. You can read about it on Wikipedia.
That's not true either. Nintendo approached Sony because it wanted to use CDs. But Nintendo didn't like that Sony had more control than Nintendo thought they should have, and effectively shafted them in favour of Philips.
/EDIT for OP:
Would rather Nintendo and MS link, as we may actually not be shafted here in Europe. Although to be fair, we did get Mario Kart, Metroid Prime and Wii Fit first. In reality, I like it the way it is. 3 fairly different choices.
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