Macbook Air strange boot-up
It seems to me that you have an issue with settings stored in NVRAM. To reset your NVRAM to default settings (it's safe to do this), power on the system and simultaneously hold down the Cmd-P-Opt-R keys until you've heard not one, but TWO boot chimes. The second is usually louder. Then let them go.

If that doesn't fix things, then you should get test the hardware. That said, this tends to fix a lot of the slow-boot issues.