ANSWER: To a degree, why God allows the devil to operate is part of the general question of the problem of evil. That is, why does an all-good, all-powerful, all-knowing God allow evil to exist ? You may find it worthwhile to read what C.S. Lewis wrote on this subject in his book, The Problem of Pain. He became an atheist when younger largely over this issue, but later on changed his mind.
A description of the devil and his revolt against God appears in Isaiah 14:12 - 15 and Ezekiel 28:12 - 19. Although it can be disputed about where the description of the kings of Babylon and Tyre begin and end compared to that of Satan himself, these verses most sensibly can be applied to him.
You were the seal of perfection, full of wisdom and perfect in beauty. 13 You were in Eden, the garden of God . . . 14 You were the anointed cherub who covers; I established you; You were on the holy mountain . . . (Ezekiel 28:12 - 14)
Ezekiel 28:15 goes on to say that Lucifer was created perfect (as far as everything that could be given a spirit being from the start, which of course does not include character!) from day one. He, however, indulged himself into thinking contrary to our Maker's will (stinking thinking) to the point where he sinned and rebelled! The devil, at one time, was one of only two cherubs who covered the very throne of the universe (Ezekiel 28:14). The question becomes why does our Maker put up with Satan's continuing evil influences on mankind (Ephesians 2:2; 2 Corinthians 4:4; Job 1 - 2)?
Although this can't be directly proven from Scripture, I believe reasonable deductions from the Biblical evidence can be made on this subject. By allowing evil to exist, our Father isn't merely trying to prove something to mankind, but also to all the created beings of the universe. Since people, or even the angels, won't necessarily take God's word for it on faith that some other way of life besides His works, He has allowed people to live and experience other ways of life. Here, this becomes a type of self-inflicted misery, but then people feel like blaming him for the evils their free choices brought upon them.
Presumably, a similar issue came up among the angels. Satan attacked the Creator and his character, and got one-third of the angels (Revelation 12:9) to side with him. These angels, like many of us humans, didn't trust God or have faith in his revealed will as a loving and unselfish Being. He is presently allowing these spirit beings (the demons) to learn, along with humanity, that disobeying his laws is not a successful or happy way to live.
God may also be allowing Satan to operate in order to help us develop holy righteous character more quickly, since Christians have to learn to resist against Satan's influence. Here the analogy would be with weight-training and the concept of resistance, only there's a spiritual principle being applied here: We develop the habits of obeying by our own free choice more solidly and permanently if we have to resist more in order to develop them.
Not only could the devil been put to death when he sinned, but also Adam and Eve. The Eternal, however, decided IN ADVANCE that if they (the humans) did sin, He would pay the price personally. Always remember, in Christianity, Jesus chose to give up his divinity and to suffer like humans from the consequences of sin and living in a world cut off from the Creator. He didn't live an easy, sheltered life in the lap of luxury. Instead, He chose to die a horribly pain and humiliating death for people (in most cases) who didn't know what He did or who actively consented to His death. We should, therefore, have faith in our Father that He knows what He is doing when He allows sin temporarily to exist in the universe as a temporary, unwelcome visitor.
God is in the process of making beings like Himself that have 100% free will (like He does) yet will also obey His holy righteous law of love 100% of the time (like He does also - compare with Matthew 5:48). This would be the main reason he allows evil to exist by deduction from the Biblical evidence.
We are not in a position to judge God and His actions, as Job did. Let's be humble and patiently await the day He'll abolish all sin and take care of the devil for all eternity.