I’m not sure that what you describe as your FB tests are signs of insecurity or else just distrust of your partner, and frankly, if you distrust your partner enough to “test” them by making up a phony Facebook account to flirt with them online, you’re already in the wrong relationship.
Why not try and choose someone who has integrity and character so you don’t have to feel like you’re being betrayed, cheated on or lied to? The minute you start “testing” your girlfriend, you should know it’s time to leave.