Sort of. Not caring about something takes the emotion away. But if you are not emotional about something you should already be relaxed.
Maybe. One would have had to feel not apathetic at all about something in the first place, though, in order to feel relaxed about becoming "indifferent" to it. I don't think that really happens very often, people actually becoming indifferent about something that they once cared much about.
Technically indifference would be lacking any feeling at all. But yes, indifference is easy on the mind.