How long have you been with your girlfriend? Has she been pregnant in the past, does she have a reason not to trust you, or the pregnancy tests? I think she's going way over the top. Being concerned and responsible is one thing, but going this far is worrying to me.
(Presuming a doctor would allow it), getting a vasectomy sounds like a huge, huge step to me. If you two aren't seriously planning on getting married I think an operation would be going way too far. (I'm also not convinced that her taking the Pill will settle her fears at all.) Having a vasectomy/her having her tubes tied would be the very final step I'd be willing to take to prevent pregnancy, even though the operation is reversable and not life-threatening, etc.
I think her fears are ungrounded, and reaching for a pregnancy test or getting a vasectomy avoids the problem, and doesn't face it. You and her should go to a doctor or a clinic together and get the doctor to discuss a range of birth control options and hopefully she can have her fears put to rest. It sounds to me like she has had a bad experience with pregnancy in the past and is terrified it will happen again.
BTW, posting this in Paranoia gave me a chuckle
