Nope, you can't pick the same champ. If it's Blind Pick, the other team can pick the same champ though.
I recommend the ff. champs. I think they're all 450 IP.
Tank: Alistar
Bruiser: Garen
Mage: Annie (offensive), Ryze (tanky offense)
AD Carry: Ashe, Cait (she's free so often)
Jungler: Nunu
Support: Soraka
Aside from Annie and Cait, they're all pretty competitive too.
I think I like rushing Sorc shoes -> DFG -> abyssal -> void staff. You should be fast enough without mobility, and unless that magic book was healing you a lot, I would just try to focus on magic pen, damage, and maybe using the ult to shield yourself. As far as I know Eve scales very well with magic pen so I max it out. Either way, once you get sorc shoes + DFG + ult you should have a big burst on anyone...I guess it maxes out to 50-60% flat health damage or whatnot.
Though I haven't played in forever so I may be a bad person to listen to.
Ahaha, I tried going Sorc->DFG->Abyss. I started with 0-3 (10-20 team scores) and ended up carrying so damn hard. I passed up on the Void Staff for now since I was already dealing true damage lulz. Enemy team pissed themselves whenever I instagib their (MR-less) tank. Bought a crapton of wards to assassinate people, got so many triple kills. Eve is so ridiculously strong right now.