There can be risk any time. An egg only is able to be fertilized within 12 hours after it is released. Typically this happens at around day 14 of her cycle (two weeks or so after her period), but that's only on a "normal" 28-day cycle. This is more of a target for people who are trying to get pregnant than those who are trying to avoid pregnancy. Sperm can survive for up to 7 days (average of 3-4 days), so if you have sex on day 7, you can still possibly get her pregnant when she ovulates on day 14. Also, some women ovulate irregularly and, although rare, it is even possible to get pregnant on your period.
In other words, don't go by ovulation to calculate your risk. People who cannot use birth control for religious reasons do, but even doing it "perfectly," there is still a high chance of pregnancy compared to other methods.
As for the mood effects of birth control pills, your girlfriend may want to go back to her gynecologist and get a different kind of pill (there are several kinds). Not all of them work the same or cause the same side-effects. There are also other types of birth control to try. I personally don't have a lot of faith in condoms because they break sometimes, they get tiny holes if they are kept in a warm environment, etc. I still use them because they are a great back-up and prevent STDs (which bc pills don't), but I personally use them with the pill for extra prevention. But maybe I'm just paranoid. It's a choice that you and your girlfriend should make together.
Good luck. If you have any other questions, feel free to post.