Hebrews 1:2
2 Hath in these last days spoken unto us by his Son, whom he hath appointed heir of all things, by whom also he made the worlds;
Hebrews 11:3
3 Through faith we understand that the worlds were framed by the word of God, so that things which are seen were not made of things which do appear.
Moses 1:33
33 And worlds without number have I created; and I also created them for mine own purpose; and by the Son I created them, which is mine Only Begotten.
In these three verses we see that through Jesus Christ, God made the worlds. This would indicate that there are more worlds than just this earth. If these worlds were populated with children of God also, then I would bet that they also went through a fall and committed evil acts in order to learn good and evil. Perhaps they too had their own Satanic character to tempt them. So to assume that Lucifer was the first being ever to do evil may be a bit presumptuous. I don't find anywhere in scripture that Lucifer was absolutely the first being ever in all eternity past to have committed evil.
Again, evil in principle exists and becomes something that is real when good exists and becomes real. They are opposites and if you create one then the other automatically exists. If I were to create a law, by nature of creating that law I have created the opposite of obeying that law. A choice, given free will, allows one to obey or disobey the law. I believe evil has existed for as long as good has existed as a potential choice as long as free will has been practiced. I also believe that since Christ has made worlds without number that the chances of those worlds also being peopled is great and that those people would have to learn good from evil to progress and thus evil is likely to have existed there as well. Perhaps even way before Lucifer started his rebellion.
Perhaps the difference between what you believe and what I believe is that you believe evil only exists when evil is chosen whereas I believe evil exists if it is possible to choose it. I think you believe that Lucifer was the first to choose evil where I believe that even though we don't have a knowledge of all eternity past, others could have chosen it before.
The existence of evil is not a problem. Getting people not to choose it is the real problem. The Problem of Evil is a philosophy which seeks to prove God doesn't exist. It is not the fact that evil exists.