You can play any way you want to and do not have to jungle. However, some champs are so good at jungling, that one usually prefers that to normal laning (Amumu, Shaco, Skarner and the like).
You are right, that a jungler means that one other player is left alone against two enemies. However, he has the protection of a turret, which is enough in the early game to survive. Plus, he levels almost twice as fast as his two enemies, so he will very soon have his ultimate available against two level 3 or 4 enemies. Of course, some champions have a hard time to stay alone in a side-lane. Make sure when you have a jungler in your team, that you have a solo-champ that can hold its own without problems.
Jungling can only theoretically be done by all champions (sure, anyone can go into the jungle and attack creeps there). However, most champions are way too slow to make it worthwhile. They would profit much more from staying in a lane. These champions should not try jungling. If you want to try jungling yourself, then select a champion, for which you have one of these jungling guides at hand. Don't just try any random champion out in the jungle.
Finally, the most important point about jungling is one that most novice players often forget. You do not jungle in order to remain in peace and level up on your own. You jungle, so you can level up and at the right moment appear in a lane for a surprise gank. Enemies have to be constantly aware that a jungler may pop out of a nearby brush all of a sudden turning their 1v1 or 2v2 lane into a 2v1 or 3v2 problem. If done right, jungling can very quickly devastate your enemies, because the jungler has the power to turn around any lane. If one of your team's lanes is having trouble, adding another champion to it will devastate the enemies and at least break their attacks. Often, you can even net a kill or two. If you're a good jungler and help out frequently with ganks, then you will soon outlevel your enemies and the game can be pretty much decided in less than 20min.
So in summary, do not doubt jungling. It is a very popular and strong strategy. However, be aware that being a good jungler is not easy. Especially, when playing together with other novices, who may not know how to deal with a jungler in their own team. I suggest, you try to get at least several dozen games of normal lane-gameplay done, before you try out jungling yourself. And then, start with practice games, so you are certain, that you will survive the early jungle farming. It is most embarassing to die to the golem when no enemy was ever present.
Execution means that you do not grant nearby enemies gold.
If you get executed, you also retain your killstreak and bounty.
If you know you are going to die (for instance, you have seen via clairvoyance that the enemy is coming and you won't be able to retreat) it is better to suffer an 'execute' to deny your opponents a successful gank. The kicker, however, is that you must not have been damaged or debuffed by an enemy champion within 10 seconds of death. (Often easier said than done)
Executions also don't show up on the "Team Kills" tally, though they do appear under a champion's deaths.
Best Answer
The League of Legends Devs recently posted a design blog about what went into developing the quests, and why.
An excerpt:
A quest triggers at random intervals (I've seen two in one match, before) and will always involve two adjacent capture points, one controlled by your team, and one controlled by the enemy.
Completing a quest, that is, capturing your target point before your enemies capture their target point, gives you a short duration 10% damage buff, and immediately deals 20 damage to the enemy nexus.