Each Role - Team Composition?

Alright, well to start this off, welcome to League! Alright, well it looks like you have a pretty decent grasp of the game to start with. So let's go over a few things.

To address the issue with ADC's - Their primary position is bot lane, along with a support. They basically farm up on minions, try to get the most kills, and carry the game by destroying objectives.

However, with League evolving as it does, with numerous patches and buffs and nerfs, the players enjoy trying new things with champions that hadn't been done before - or at least, something rarely done. ADC is not necessarily a midlaner, as midlane is usually for assassins, mages, and casters, but people have recently noticed that some ADC's (Varus, for example) can be played midlane for good results. Of course, this isn't set in stone, and you can play most any champion anywhere, they usually have a position they play better than others.

People may play champions in other roles, but the stereotypical team comp is:

Top - Tank/Bruiser (Or anyone that can bring something to the team and doesn't really fit in other lanes, like Teemo) Jungle - Similar to Top, except a wider variety. Junglers have sustain, and a decent clear speed, and high gank potential. Mid - Usually reserved for Mages, Assassins, and Casters. These go for kills and try to get fed and carry. These usually are AP, with a few exceptions (Yasuo, Zed, Talon, etc). ADC - Bot lane. Farms minions and tries to get kills to carry the team. Always an AD champion. Essential for killing turrets. Support - Bot lane. Champions with a lot of utility/poke/crowd control. These do not kill minions, poke the enemy laners, and depending on the champion, engage for the ADC.

To address the part of Ahri, she is usually a mid lane. She doesn't really do well in other lanes, although I've seen an Ahri try support. To be honest, you're pretty much stuck mid with her.

And for why people always want mid - It's the best lane, obviously. (of course no bias, because i'm a mid main ;) ) but seriously, people like to go mid (And remember, in order of popularity, it goes Mid>Top>Jungle>ADC>Support) because it's the easiest lane to get kills. People enjoy getting kills for obvious reasons, and you're obviously going to get more kills as an Assassin mid, than say a Tank top.

If you have any other questions, feel free to ask, and I'll answer them to the best of my knowledge :)

Top is usually for tanks, but there are many champions that can go top lane as well, such as Heimerdinger, Vladimir, Riven, Jax, etc.

Middle lane: This person will usually go AP, but there are also AD mid laners such as Zed.

Jungler: Can be a tank or a bruiser (high attack but low/mid defence).

ADC: Stands for attack damage carry. Logically, they build AD. Their sole purpose is to farm and win lane so that they can become relevant later in the game.

Support: Sole purpose is to babysit the ADC and also have utility in teamfights. These are generally tanks because they build items that, while also giving them utility abilities, have defence stats (Locket of Iron Solari, Frozen Heart, Randuins Omen, Righteous Glory).

Terms

AD: Attack Damage, which is damage based off auto attacks. AP: Ability power or Magic damage, which is damage based off abilties.

So you usually want a balanced team that attacks on several angles. If your team is literally all AD, then your opponents can counter you very easily by buying thornmails and Randuins, If you do not have one or two tanks, then your teamfighting ability will be severely affected. Here are some healthy team comps:

Top, AP (Examples: Rumble and Maokai, the latter having the bonus of also being a tank) Mid, AP (Cassiopeia, Ahri, Leblanc, Azir, Annie, Viktor, etc) Jungle, fighter (Vi, Lee sin, Kha Zix) Bottom Support building tank, can be either AP (Morgana, for example) or AD. ADC, always AD.

or

Same as above except top builds ADC. Sure they can build Banshees or Locket to counter your midlaner, but they also have the AD damage coming in. Preferable that the mid snowballs so that they have to counter it.

or

Top AD fighter (Irelia, Jax, Hecarim) Mid AP ADC Support (AP or AD) Tank Jungler (Shyvana, Sejuani, Hecarim).

If your support builds tank, then you have a nice 2 tanks!

As for Ahri, she is not very good in other positions. You can try her as support or in top lane, but your opponents can always take advantage of that. I suggest you try the jungle, because it is awesome:

1. You will have no one harassing you will you farm (exceptions apply)
2. It is always different
3. You can build a variety of champions (Tank or figher, sometimes you can even go AP with champions like Diana)
4. You get to play a role in all lanes.
5. Can carry games too.
6. You dont have to worry about last hitting.

Normally you can have crappy team comps at low levels because most times someone will just get extremely fed on either side. But you have to play good champs in ranked and later levels.