Yes you can buy up to 3-day passes from the gate. If you want more than three days (such as a week long pass) you can ONLY purchase those from the USC Transportation Services Office.
This is true as long as you have a valid USC parking permit (includes all off campus lots and Parking Center passes). With a permit you can park anywhere on campus (except for reserved spaces and loading/ unloading spots, please obey all parking rules and regulations) after 5pm and anytime during the weekend.
All of the gates are open during USC home football games, so enter the gate that is most convenient for your travel. Regular permit holders can park where their permits allow, but guests must purchase parking at the gate.
Yes you can.