You don't need another paid online account, she just needs her own gamer profile on the console rather than playing as guest. You only need one PS plus account per console. When she makes her EA account, make sure it's the same gamer tag and email address as the PS profile she makes. It's pretty straightforward but I can't give quick instructions because it's been quite a while since I did it.
You can all four play ops or any multiplayer mode together. You can play the multiplayer modes online against other players or a "private" game against AI's. Each player can choose to be a plant or a zombie, so you could all play together against an AI team or split up on to opposite teams. Or you can play a multiplayer mode with just the four of you, but generally speaking four players aren't enough to make those game modes worthwhile. You can also just play around together in the back yard as any combination of plants or zombies you wish.