If god cast out the angel lucifer into the eternal lake of fire which he will reside forever and ever why would he be allowed to be in the human form on earth? Why shouldn't he still be in hell? Is he allowed to be out and let loose on the earth for some reason?

Yes he is because man has free will and would rather follow evil.
The Bible calls the demons the principalities and powers of the air.
Revelations explains about when he gets bound for eternity.
I am sure someone here can explain this better.

How come he is not burning in hell where he belongs?

because of adams sin, sin nature is what everyone has at the moment of birth. The time has not yet come for lucifer or satan to be in hell

Lucifer (Satan, the devil) is a fallen angel, so could appear to someone just as any righteous angel could. He is not in Hell, has never been in Hell, but will be held in prison during the Millennium and then be released for a "short season" to try the inhabitants of Earth, and then be thrown into the Lake of Fire at the time of the Great White Throne judgment. He is allowed certain freedom for now to give all of us the opportunity to exercise our freedom of choice.