Several possibilities - more background about how/when the behavior started would be helpful.
Successful boot to safe mode means corrupt HD or System files. A boot to repair disk utility may be able to fix that - or - if it can't you need a new HD. A single user mode boot may also fix some corrupt system problems.
That you can't boot to an external device could be the type of device, and/or the system on that device. If you can Target boot the problem machine you may be able to diagnose/repair the disk problem or clean reinstall the system.
If this answer is acceptable please remember to return and mark it.
Do you know if this machine was affected by liquid spill ? If so the top case could be damaged and acting strangely. I would begin the troubleshooting by booting the MB with an installation disk and check if the issue is a software one. Next I would remove the bottom cover and check the keyboard connection to the logic board. Inspect carefully the logic board for any sign of stains. Habitually I remove the logic board to inspect the bottom side also. Another step is to disconnect the keyboard cable and start the machine by bridging the two power pads located near the keyboard socket. If the machine is working ok in this setup than the top case is the culprit.