Who is Satan? Why is he allowed to exist? What do we know about him?

Is Lucifer and Satan the same? The Bible says Satan is from the beginning a liar, from the day of his creation, but Lucifer is not like that, he was a holy cherub till the day he was found guilty.

If God does not allow sin to enter into Heaven then why is Satan allowed access to accuse the brethren?

I understand at one time Satan was an angel in Heaven, the most beautiful. I have often wondered where his sin derived from to want to "take over". As I understand it, he was the one to start "tempting" to cause Eve to sin, but what caused his sin to become what it led to?

What facts do we know about the fall of Satan? Did he cause other angels to rebel with him? Were the angels deceived or did they know what they were doing?

Can you explain the woman being hid in Revelation 12 and Israel's part in the tribulation?

How fast is to and fro? Is it instantaneous or is it hours long when the devil went to and fro upon the earth in Job chapter one? Was it in seconds or did it take all day?

The devil was certainly created before the creation of the world since the Bible says the devil was

Is it scriptural for churches to plant more churches? Is a church unscriptural if they were started because of a split? What is a church arm?

What is the correlation/connection between water baptism and church membership? I see where Acts 2:41 references both baptism and adding to the church in the same context, but have not found where baptism is clearly required for church membership.