Jungle suggestions (I don't play Jarvan or Naut enough to give you any insider-infos):
Since you don't need to worry about bans, Nunu is the creme de la creme.
Current junglers you will see in competitive play:
Nunu : Would play everytime if unbanned, too OP. Focus on counterjungling and objectives to win games.
Zac : AP Tank that deals % damage and has revive passive, very disruptive in team fights with Ult.
Elise : AP Tank that deals % damage, decent ganks with long range dives & stun, can avoid damage & turrets with rappel.
Jarvan IV : Very good but his ult can be mitigated easily when facing a team that has a lot of gap-closing champions (champs that can easily exit his ult). Good vs Udyr and Singed players specifically.
Nautilus : Annoying as F*ck, too OP at higher Elo where teams follow up on his CC more often. Probably annoying to play at lower Elo if teammates don't take advantage of his CC. He lacks damage.
Nasus : Annoying as F*ck. Builds all tank and still does tons-of-damage at late game. Can carry games if they go late.
Shen : Hard to play well, low win rate, low damage, global ult needs to be utilized well, taunts can change games if teammates are able to follow-up.
Lee Sin : High skill cap, falls off in late game a bit, can carry games, very mobile. Good Lee Sins are annoying as F*ck, bad Lee Sins get steamrolled.
Nocturne : Global ult, works well with a shen or TF on the team, very good ganks, spellshield works great vs AP teams, does decent damage but usually builds tankish.
Cho Gath : AP Tank with an Execute ability, TheOddOne helped TSM win the Spring split with his masterful Cho Gath play. All around good champ that can carry games.
Normally permabanned in Solo-Q:
Amumu : His ultimate wins games, landing bandages will fuck up noobs.
Hecarim : High speed champ that doesn't care about wards or anything, ganks too good.
Nunu : See above
Some top-laners that can jungle:
Jayce : Jayce is just OP atm, being changed soon
Tryndamere : Needs farm to get good, just like with top lane, he might get farm slower but once he has his core items he will wreck faces as usual
Jax : Gap Closer+Stun+High Damage, nuff said
Riven : Gap Closer+Stun+High Damage+Execute Ult, nuff said
Aatrox : Gap Closer+Knockup+Slow+High Sustain, good for counterjungling, ganking, and soloing dragons
Darius : Hook+Execute, landing the hook is key to a good roaming Darius
Kha Zix : Stealthy Assassin, lil too squishy, the surprise attacks are key.
Some mid-laners that can jungle:
Evelynn : Stealthy Assassin that specializes in surpise attacks, ignores wards or forces enemy to invest in pinks, somewhat tanky with Ult and has high mobility.
Diana : AP Tanky champ with great gap closer, a cc and decent AoE damage. Good champ overall - better as a Mid, though (needs farm).
Fiddlesticks : OP Champ, Fear lasts for days. Really OP kit. Specializes in the surprise ganks and usually wins most 1v1s due to life drain.
Master Yi : Getting changed soon, high mobility+high damage+skill resets, lacks CC but his chase potential makes up for it.
Others to consider (some of these are viable as top or mid, too):
Volibear : Mobile+Flip+Slow+Execute+Tank+Regen Passive = Still OP imo. (Consider as a top laner, too)
Udyr : Versatile kit but suceptible to being kited.
Vi : Still good IMO, considered more of a "carry" jungler. Does pretty good damage with good diving potential.
Mundo : Beefy, can't really carry, funny champ, cleavers for days
Xin Zhao : Win Now was considered OP, might have been nerfed or fallen out of favor recently, still decent imo. Builds more toward damage, has gap closer+knockup. OP for ganks.
Malphite : Was 100% banned in Solo-Q, not much anymore, nothing really changed. Tank with great ultimate and a slow. Counters most melee champs.
Maokai : Still good imo, Tanky, AP damage, gap closer.
Rammus : Taunt still annoying. Extremely fast+taunt = OP Ganks. Builds tanky, good vs AD
Sejuani : Tanky AP with tons of CC, good initiates with Ult. A good ult can win games.
Shaco : High Skill Cap, Good Shacos can carry games, Bad Shacos are dead weight. Can be annoying as F*ck. Requires team to invest in pinks. Normally farms enemy jungler.
Shyvana : Was recently changed, kills Objectives insanely fast (drag/baron), still lacks in the ganking department. Dragon form is required for good ganks. Consider taking Exhaust for CC otherwise you have none.
Skarner : Ult-Bot. Kinda useless except for his Ultimate. His ult can win games, though.
Trundle : Tanky and annoying as F*ck. Too fast, decent damage, too much cc, great ultimate
Warwick : Great against noobs. Ult gets countered with QSS but most noobs wont buy. Fast champ, great ganks, great sustain, can do good damage when fed. Ult can win games.