How is a shorter distance to first not quantifiable?
American football is the only ball game where the ball is not ball-shaped.
Halfball is the only ball game where half of a ball is used.
Hacky Sack (or footbag) is the only ball game where the ball is normally deflated but round in shape (as compared to bean bags.)
Snooker uses the most balls (22) of any ball game with a limited (vs. pachinko) number of regular starting balls.
That’s no better than your last effort of American exceptionalism.
Rugby union, rugby league, Australian Football all use (different) elipsoid shaped balls.
Ultimate does utilize observers at least at some of the tournaments. The collegiate finals were here one year and there were observers on the field for that. If you watch highlights from the USA vs. Japan world championship game you can see some people dressed in gray that I believe are observers.
This is seen as a serious foul in Rugby Union, which usually results in a yellow card (and hence ten minutes off the pitch) for the offending player.
This is true of croquet (though this is perhaps not a serious sport in the context of this thread), and the latter is true of snooker, even at World Championship level.
This was a football (soccer) match, not Aussie rules, where the tiebreak rules declared that if the game ended in a draw, the result should be settled by penalty shoot-out. A penalty shoot-out win was deemed a 2-0 win under the rules of the competition. In one game, Barbados needed a victory by 2 clear goals, but were leading 2-1 with only a few minutes left. The rest of the story is predictable and is well-told on snopes.
There is also a hill (complete with flagpole) at Minute Maid Park, home of Major League Baseball’s Houston Astros.
I’m not certain, but is baseball the only sport in which it is allowed for the playing field to be not entirely level?
ETA: It took about half a second after hitting Submit to realize the answer to the preceding question is no.
Golf
In baseball, a non level field is a requirement if you include the pitcher’s mound. It’s not unique in this, however, Crown Green Bowling is played on a convex pitch (with asymmetric balls).
You can add Canadian football to that list.
Also, I’m thinking tennis is the only sport where a player wins one unit (in tennis, point/game/set) at a time, but the winner must win by two or more units.
I think volleyball must also be won by two or more points, mustn’t it?
and table tennis, squash …
Okay, let’s ignore that about tennis, then. Hey, I’m easy. 
I think that a reasonable definition of “offense” is “the side actively trying to score points” while defense is “the side trying to prevent points from being scored”. By that definition, the batting team is on offense in cricket.
The OP mentioned Hockey, but no one’s mentioned that it played in three halves, no quarters, er whatever. Any other sport have this?
But the score in cricket isn’t measured in points, so the premise is wrong.
Substitute either “runs” or “wickets” into that definition and you get different outcomes.
Cases vary. In some situations, the batting side is aggressively trying to score runs. In others, they are defensively trying to avoid losing wickets. (An example would be a team 150 runs behind in the last half-hour of a Test - a time-limited international - with seven men out. There is no hope of scoring the runs, but a reasonable hope of lasting out time.) And sometimes both can be true at once - a side can have both a chance of winning and a real risk of losing, which makes for extremely interesting cricket. Indeed, sometimes a fielding side will give up some easy runs to encourage the batting side to go for it, at increased risk of losing.
Well, tennis in most tournaments is played in three sets, but it’s not quite the same.
It’s best of 3, actually, so it’s really not the same, but similar.
That “points” are called “runs” doesn’t change anything.
I realize that wickets are important, but at the end of the game, it is runs that matter. Yes there are cases (as Malacandra notes above) where a team can find themselves merely trying to defend wickets, but such a situation arises because they are behind in runs.
Look at it this way. Take a 50-over ODI where the team batting first ends their innings at 250-10. The next team comes up and, while they very successfully defend their wicket, end the day at a mere 180-3. Who wins, the team with the most wickets, or the team with the most runs?
That would be three periods.
Seriously, “halves”? I would have understood “thirds”, but “three halves”? Are you one of those people who always gives 110%? 
In answer to your last question, the team with the most runs wins. But that is because it is a limited-overs game, whereas Malacandra was referring to situations that occur in limited-time games (usually 4 days or 5 days), where the batting team can be well behind in runs but play for a draw by batting defensively and conserving wickets. If the bowling side cannot get the batting side out, the match is drawn.
I understand that, but again, the reason that they are playing for a draw is because they are behind in runs. In the end, it is the runs that matter, and the number of wickets taken is ultimately not a measure of anything.
Although, going back to the OP, cricket may be unique in allowing a draw when one team is losing.