Prim was in the Capitol because there was a hovercraft sent out to help the wounded rebels because they needed as many people as possible to help the rebels
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Gale didn't kill Prim. It was actually that Coin lady. Which at first I thought it was Snow.
He bombed the courtyard where Prim was taking care of the wounded in order to ensure that all of the remaining Capitol supporters were dead. He did not know that Prim was in the courtyard trying to save lives. Coin couldn't control Katniss. She orders Prim into the bomb zone to take care of the wounded. Knowing she will be killed. She has to break Kat's will and mind. Killing Prim does this. Kat finds out it was coin who ordered Prim into the danger. She kills coin with an arrow in the end while Snow laughs. The then drops dead from all the poisons he's been taking.
Prim dies on page 406
Prim is the only person that Katniss knows she loves. She tries to protect Prim in every way she can.
President Snow ordered the bomb that killed Prim.