Do you have any way to tell when you have GPS lock ? Is it possible it is still acquiring satelites ?
After 6 minutes, he is up to 9 satellites. I don't think the GPS reading is the problem.
The problem lies with his pressure device. His pressure reading is fluctuating wildly between 0 and 200 kPa. In other words, he is reading complete junk. His I2C connection is not reading anything useful at all.
And then, his alleged barometric altitude is showing no correlation to the alleged pressure readings at all, when in fact there should be a perfect although non-linear correlation, because the altitude is determined directly by the pressure. The OP has not said whether he is using the altitude calculated by the device, or whether he is calculating the altitude from the pressure on the arduino. Either way, it's 100% junk.
The whole question of it taking 10 minutes to "warm up" or "settle down" to a stable reading, is effectively a complete red herring, because all of the supposed readings from the MPL3115A2 are complete junk.