Well, with the time and draw thing; everyone wants to win in chess. When you had a few seconds on your clock he knew he was going to win, why would he want to accept a draw when he knew that. I would of done the same if I was in a lost position. Its just the game of Chess, its ok.
Good sportsmanship?
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To the OP, the short answer is: No
Yeah, there's nothing wrong with taking advantage of your opponent's time trouble in a rapid time control game of chess, and no one is obligated to a rematch.
Moreover, drawing a QvR endgame for the side with the Queen should have been a fairly easy and straight forward task, even with only 2 mins on the clock.
In once of my recent live games, I reached a winning position, Q vs. R with no pawns on the board. However, in reaching that position, I used much more time than my opponent, so that I only had about 2 minutes left to convert the win. With just seconds remaining on my clock I offered a draw but my opponent refused it, and when I lost on time he refused my rematch request. Is that bad sportsmanship on his part?