The Karhal is the ruler. And yes, it is the highest class, composed of only one individual.
This man is "chosen" only indirectly. He still fights, like everyone else, to maintain his position at the yearly games. However, like a Chinese Olympics, the winner is semi-predetermined beforehand by the political maneuvers and promises of the highest and most influential men of the kingdom.
For example, if I wanted to be the Karhal, I would have to make rather significant promises to my close competition to persuade (ahem...bribe) them to allow me to win our duels. However, most of the best warriors are running deals with multiple other warriors, and since most of those deals are strictly confidential, I wouldn't know if my promise was not as big as my neighbor's promise to this guy here that I'm trying to convince to 'let' me win.
There is usually some general consensus among the best warriors as to who would best fill the position of Karhal, and it is usually this man who ends up with the position but this doesn't prevent side-deals and some individuals vying whole-heartedly for the position themselves.
For many warriors, the position would be more of a bother than a bonus, though your word is ultimate when it comes to national campaigns and things of that nature, and you are supreme judge of the land. The position of general is to most of these men more attractive than that of Karhal.
Good questions about using the tests as a cover for murder. The penalty for killing your opponent during the games is death. There are several guards present at any contest, and if one opponent kills the other, they execute the offender forthwith, even if it was an accident.
All contests are held with fully functional weapons (e.g. no blunted swords), and wounding your opponent is legal.
If a contestant wounds his opponent to such a degree that he dies more than seven days later, there is no penalty, because they believe that by that time a disease secondary to the wound (ie. infection) has most likely caused the death, and not the wound(s) it/themselves.
However, a wound resulting in death less than seven days later requires the administration of the same wound (or as close an approximation as possible) upon the offender's body. (This wound is administered by a non-influenced third party.)
If a wound results in death on the seventh day, a relative of the deceased may issue a challenge to a duel with the offender, which the offender cannot refuse (according to societal standards; technically he could refuse, but he'd probably end up getting killed).
In the case of multiple wounds received in different duels adding up to death less than seven days later, each offender receives the wound that he gave the deceased individual, or as near an approximation as possible.
In the case of multiple wounds received in different duels adding up to death seven or more days later, there is no penalty.
So wounding your opponent is kinda risky. The point of the tests/duels are not to defeat your opponent so much as to demonstrate superior skill (at which time the duel ends).
This demonstration of superior skill in weapon handling is judged on the basis of who controls the fight more of the time, number of hits (light armor is optional, most prefer to wear it), area of hits (head hit is automatic win - but also potentially dangerous to attempt because of wounding/killing rules), and other points.
The judges are those in the scribe/strategy class. They are not usually themselves proficient fighters, but must demonstrate complete mastery of combat strategy, precise knowledge of weapon uses and skills - basically, they're a very specialized peanut gallery of guys who have usually been watching and judging duels for years. Their word is law for the duel that they are judging.
Hope that answered your questions!

Thanks for asking!
-Ka