Format the card via the DJI Go app while it is in the Phantom, not your computer. It can sometimes make a difference. I know it does with digital cameras. If you can't, it should necessarily be a deal breaker. But it is better for format in the device (Phantom).
Again, make sure you download the .bin file (not the .zip file) to the SD card. The .bin file needs to be in the root directory (which means that it's not in a folder on the card, but simply loose on the card). Also, make sure you've downloaded the P3A firmware and not the P3P firmware, or P2 or some other Phantom. And make sure you have your controller turned off while you're updating firmware via an SD card. While you're at it, also remove the camera transport bracket. The camera needs to be able to move about during a firmware upgrade. You may hear some noise from the gimbal motors during an update.
Firmware can take up to a half hour to install. So have your battery charged to at least 50 percent. You should hear a series of beeps as it installs. The there is an LED on the front of the gimbal, above the camera lens. It should flash green and red while firmware is installing. It will turn green when installation is complete. Disregard the LEDs on the motor arms.
I don't know of any way to update P3 firmware connecting to a computer. With the P2, you connect to the computer to update firmware, but you also use an app for that.