Because that might as well be a death sentence. The ship's power would eventually fail, and there's no guarantee anyone would find them (and would you want them to?). Kirk's decision was logical. Give them a new start on some remote backwater and let them see what they can do. Also, let's not forget it was Khan's choice, not a sentence.
Absolutely. TWOK ret cons it as having been "exile", something which was apparently forced upon Khan by Captain Kirk... but our only account of that in the movie is Khan's own, and he's had decades of bitterly twisting the facts inside his own head. Kirk became an outlet for Khan's anger over his little colony going wrong, Marla's death, everything that Khan the supposed superhuman was utterly helpless to prevent. It's misappropriated blame on Khan's part, really.
In reality, the decision made to put them on Ceti Alpha V in "Space Seed" was presented as being much more magnanimous and logical than that. And as you say, Khan in the episode sees it more as an opportunity for a new life, not as an exile being imposed upon him.
Khan did had a legitimate grievance with regards to no one from the Federation coming by to check on them though.
Did Kirk not tell Starfleet what he had done?