I believe that most of the meaningful questions are more nuanced than that.
Nope. Quantum physics demonstrates that two things can be true at the same time in the physical world and I believe that there are also philosophical ideas which can similarly hold within them inherent contradictions. We call this a paradox and both the Buddah and Confucious detail many such intellectual games.
Listen to Zen master Alan Watts for some interesting ideas on this sort of thing .
No, not everything is objective. Granted, sometimes we take this too far and make objective truths out to be subjective or relative when they really aren't. Such as this bullshit that everyone's belief system is true in it's own way. Nope. Bullshit. haha