Is belief a choice? Why or why not?
I suppose it depends upon how you define choice. We may believe something to be true, simply because it is an assumed truth because it's how we were taught or programmed to think through our families and culture. But is that really a choice?
If once given information that may challenges those assumptions though, then I'd say choice comes online. We typically choose to stick with the assumptions, at first at least. Or then we choose to question them. Once there is an alternative view offered, then I'd say we are choosing. Yes.
Is lack of belief a choice? Why or why not?
It depends upon if by "lack of belief", you mean simply a passive ignorance of information, or if you mean it in the sense of an active "lack of belief" in some specific question you are fully aware of as a question.
If it's the former, just plain unawareness of something, then of course not. That' not a choice. If it's the latter sense of a "lack of a positive belief" in something you are aware off, then that is a choice. A negative belief in a known proportion is still a belief, in that it was given a conscious consideration and a choice of either a positive or negative belief. That's clearly a choice.
So yes, if someone has never even heard of God, then their "lack of belief" in God is simply an absence of awareness of the question. That's not a choice. But those who have heard of the idea of God and considered it, and then don't believe it, that is a choice. It is a belief.