who decides which questions to post on twitter or is this an automated process for questions that gain a certain amount of votes in a certain amount of time? Is it linked to HNQ status?
It is an automated process. There is a "hotness" algorithm, which requires among other things a positive score. I don't think the full algorithm has been disclosed, so it's unclear whether it's tied directly to HNQ status. Empirically, it seems that all questions are tweeted before they become HNQs (but not all tweeted questions go on to become HNQs).
And why post questions on twitter at all? Will this not lead to a voting bias for those questions (in the sense that those questions will receive more votes this way than they normally would, making them appear more important than other questions)? Or are they in fact more "valuable" questions and thus worth posting on twitter?
Agree that this leads to a hugely multimodal distribution: questions that get tweeted or become HNQs can easily reach 1K views, whereas questions that do not almost never get more than 500 views. Though of course, the most interesting questions are the ones that are promoted, so this is partially correlation rather than causation. But to the extent that the number of views (and votes) correlates with question quality at all, it is certainly not linear.
Perhaps most concerningly, the first few hours are critical: if a question starts out bad but is then edited into an amazing question within a few hours, it is probably too late for the question to get promoted. In some cases, it will even be stuck with a hugely negative score that it no longer deserves.
But overall, this is a feature not a bug; by making the most interesting questions more visible, we can attract new users to the site.