Some questions.
In below 1200 elo... should I worry about counters or stick with my favorites?

What I'd like from you is to tell me when my champs would be ideal in regards to lane opponent and team comps, etc.

Top:
Jax
Teemo
Cho

Mid:
Morde
Eve
Swain (viable? Friend says not to waste time)
Ziggs

Jungle:
Mundo
Alistar
Rammus
Skarner/Lee Sin (both are good, I'm just horrible with them)

ADC:
Twitch (by far my favorite char)
Ashe
Tristana

Support:
Janna
Sona
Nunu
Kayle

Now, obviously I'm not good with all of them, I put them in order of my comfort and ability.

Any advice would be appreciated.

Just wish I found my 'Twitch' of every role.

Well first off if you're concerned about how fast your elo is dropping I suggest you stop playing ranked for a little bit. That being said, your champion collection is not stellar as far as counter picking.

While winning a lane does ultimately come down to skill, a good counter is helpful.

Ideally what you want to do is get where you're comfortable with >6 champions for at least 3 positions. This allows you to counter while still playing a champion you're good with.

It basically sounds like your problem is that you want to jump into ranked too soon. It's not a good idea, trust me, I did it like everyone. Honestly you're just hurting yourself and the more experienced players at your elo.

So imo go play a lot of draft.

Here are some suggestions for champs:

Top:
Irelia
Riven

Mid:
Ahri
Fizz
Ryze

Jungle:
Get comfortable with Lee Sin, he's ****ing amazing
Shyvana

ADC:
Ezreal
Miss Fortune
Graves

Support:
Leona
Alistar (usually banned, good to have though)
Nunu

Top: Yorick, Cho'gath, Teemo, Kayle
Mid: Galio, Mordekaiser, Diana*
Jungle: Cho'gath, Lee Sin, Jarvan IV, Malphite*, Amumu*
ADC: Vayne, Caitlyn, Ezreal, Graves
Support: Taric, Leona, Nunu, Sona, Alistar*


*'s are often banned.

The reason I list the champions I listed is because all of them can carry and all of them are viable at all stages of the game... They also either have high win %'s or they are simply difficult to counter in lane.

Yorick and Kayle top, for example, counter most solo-tops.
Galio is to mid lane what Yorick is to top lane.
Vayne snowballs hard.. Caitlyn out-pokes just about everyone.. Ezreal and Graves are great bursty ADC's who can secure early kills..
Taric, Leona, Nunu, Sona, And Alistar all have great lane presence, can get their ADC's fed, and transition well into late game.

Don't take this the wrong way, but what I'm hearing is:
Don't play champs you enjoy, play the FoTM champs?

Well, the first thing you're doing wrong is entering ranked games too early. You have no experience with draft games, too few wins, you're ruining the game for yourself and everyone else. Read guides, watch streams and practice in normal drafts to solve this issue.

Secondly, some champions you play are NOT viable at all. An example would be Kayle support, or Rammus jungle, or any of the mids you play.

Top:
Jax
Teemo
Cho

Mid:
Diana
Orianna
Morgana
Ahri

Jungle:
Mundo
Skarner/Lee Sin (both are good, I'm just horrible with them)
Shyvana

ADC:
Twitch (by far my favorite char)
Ezreal
Graves

Support:
Taric
Sona
Nunu


This would be a more appropriate list

http://www.youtube.com/watch?v=5MTO2a7ZFcc

Elo, Elo hell, and how it all works and why you are where you are

http://www.youtube.com/watch?v=l9Jz4Jo4em0&list=UL

How one champ can carry you up the Elo ladder

http://www.youtube.com/watch?v=JbIa20uFblg

Some of the problems with playing only one champ and what to do about it.