The only requirement to play ranked is that you need to have a lvl 30 account and have access to at least 16 champs (most of you already knew this), and Riot made these the only requirements intentionally.
Now, some people will say things like, "You need to have at least 2 champions to fill every role" or "You need to be able to get at least X% of the cs by minute Y." However, don't let them indimidate you with their raging. Ranked is a game mode where you ought to try your best to win, but it doesn't mean you need to have mastered Z% of the game mechanics or champions before you can even try it. Ranked is there so that people have a visible record of their accomplishments, partially as a way to promote players to be competitive.
For example, there is an EU gamer whose name is Malmortius that tried to get to 2k elo by playing only Poppy in season 1 (sadly, he didn't quite make his goal, but he got pretty far). Now, while he may have "auto-locked" Poppy, he was willing to play her as a support, junger, AD carry, or solo top based on the team's needs. Now, some people were mad at him for choosing only Poppy, saying that he was gimping his team, but in reality he wasn't because he kept his sportsmanship throughout his matches and did his best to help his team win, and his elo would reflect the constraints he had placed on himself.
All I'm trying to say is this: Ranked is for those who want to try their best to win and get visible stats as a result, but it doesn't mean you need to meet anything beyond the basic requirements Riot has placed for those who want to play Ranked. You can still have fun and play weird builds like AP Yi or AP Tristana or do something out of the ordinary such as double jungle as long as you make an effort to win without throwing the game.
Edit: I got a bit off track and thought I might give you a bit of feedback on the champions you listed:
Garen: fantastic at lower elo as top solo (not as good past 1400)
Amumu: good jungler at almost all elo as long as your team protects your start (get dat blue!)
Veigar: good AP mid in skilled hands, but careful of being countered by Morg/Kass
Karthus: solid AP mid all-around, and can help the team snowball (especially at lower elo)
Sivir: decent AD carry that offers more utility in terms of pushing and AOE damage than most (less single-target damage)
Warwick: outclassed by almost all common junglers, even Nunu (Trundle is a really strong jungler if you're looking for one. I use him a lot)
Poppy: Very unique, but takes a LOT of skill to truly play effectively (great at stomping lower elo)
Nunu: decent support jungler, but he needs a small buff from Riot to compensate for the new jungle (again, I'd recommend getting another jungler for higher elo, once you have the IP/RP that is)
If you want to get a support or two before going into ranked (again, entirely up to you, despite what the haters and naysayers shout): Soraka is cheap and Sona/Janna are fantastic right now. I hope this helps you out.