While your logic is fine, the usual approach to the logic would be to compute the expected return of a roll, not base it off the expected number of throws to a win.
In one roll you'd have a 5/6 chance of losing \$5 and a 1/6 chance of gaining \$21-\$5. From that you can compute the expected winnings per roll (which is indeed negative).
The question about "probability of making money from this game" is not really answerable, since previously in the question it was made clear that we're not simply pondering rolling exactly once (...each time you roll both dice) but does not specify how many times we roll/what our stopping criterion may be. You could compute the probability of making money from a single roll but that's not what it seems to be asking.
There's an implied assumption in the question that is not justified -- that nobody would play a game which had negative expected return.
Often elementary probability questions rely on this assumption that the only possible choice is to base decisions of expected value but this is emphatically not the case. It is at odds with what we observe -- in fact many people are not only content to knowingly make negative-expectation bets -- many approach such a prospect with eagerness.
If such a decision was based on amount of enjoyment derived per dollar spent compared with alternatives for spending the same money in other ways (taking a friend out to dinner, or going to see a movie, or whatever else you spend money on), then it may be perfectly rational to take a losing gamble.
I've watched many a friend or relative gamble a few dollars per week on a lottery (with a substantial negative expectation) and they certainly seem to get enjoyment from watching the draw take place. To my mind the length of enjoyment is considerably too brief for the dollars paid compared to alternatives, but that's their judgement to make, not mine -- and in any case perhaps they're enjoying the times before the draw as well. [Problems can arise when it's not the case that such a weighing of alternatives is made, since we're often not particularly rational in our behavior and may continue to make choices we can see are poor.]
By the same token, a risk-averse person may be perfectly rational in avoiding a one-off gamble with positive expectation, because they dislike the downside more than they enjoy the upside.