Runes are based on Early Game, because Caitlyn is a "Killing Machine" but only when she farms a lot. Anyway, this runes deals Caitlyn the possibility to make heavy damage from the begin of the game.
If you think that you need some armor, you can pick Defense per LvL or Magic Resist per LvL.
Other choice are Critical runes, that focus on the whole game criticals, but don't give you as many damage as this ones for Early Game.

Items in Cailtyn are really different in each situation. The order I think is the best for Caitlyn, but you can change your first object (Boots+Potions) for a Doran Sword in case you have a support that heals you from begin. The important thing is to try to stay until you can buy BF Sword, last boots and the Vampiric Scepter for not loosing LvL and farm.

About why Infinity Edge before The BloodThirster is easy. You dont need that much life steal and extra critical damage is perfect for Midgame and the extra 20 pts of damage that BloodThirster gives you when is full of charges is covered by the damage runes you have.

Then Phantom Dancers are absolutelly necesary for Caitlyn. They give her the extra movement speed and the Att. Speed so she can 1vs1 stealing HP form Midgame. You can sell at the end the boots for another Phantom Dancer, which covers the loosing of speed from boots and gives you more Att. Speed.

Frozen Mallet aren't necesary, is just for using against Teemo or characters that runes more than your team and dont have enough stuns for snare him. Instead of it, you can buy another BloodThirster so you will heal yourself when you get low HP from a battle just by hitting a few creeps.

Just last hitting for avoid ganks and enemys combos. Try to harras your enemys when you can, but focus on farming from the first moment. When you have the chance, try to push top and bot for farming with no enemys arround you.