In a relationship, D/s or otherwise, who is responsible when it comes to trust? Does trust need to be earned, or is it something that should be freely given?

I believe that trust is given. If you want to be considered trustworthy, you must trust others. It is the responsibility of all parties to make trust the first offering to the new relationship.

If there is mistrust, who is at fault? The person who has the feelings of doubt, or the one who isn't being trusted?

It depends. If you are projecting your hurt from previous relationships onto someone who hasn't violated your trust, then you are at fault. Sure, we've all been burned. The trick is not to extrapolate from the actions of people in our past to everyone else in our future.

What if there aren't any reasons to be mistrustful, but it is just a result of insecurity? How is this best dealt with?

With open and honest communication. Identify what is triggering the mistrust and share your insecurity with your partner. You can also just get over yourself. If you want to be more trusted, you need to be more focused on the needs of others.

What happens when blind trust is provided, but undeserved?

Just because I think trust should be given, doesn't mean I believe in blindly trusting the undeserving. As my Father was fond of saying, "Trust but verify."

Is it enough to just say "trust me?"

I think that strength grows from vulnerability. Just saying 'trust me' implies that you are without faults or doubts. It is better to admit you don’t know all the answers and be receptive to the opinions and ideas of others. They'll trust you more for having done so.

More specific to a D/s context, what if a Dom states that if his sub does not trust him, she will be released?

If the Dom/me has clearly stated their expectations and the sub is unable to comply, then they should be released. Remember, one way to encourage trust is to be dependable. Do what you say you will do.