Do Protestants believe Jesus and Satan are brothers?

Do Protestants believe Jesus and Lucifer are brothers? Emphatically No.

Why we don't believe this is because the Holy Scriptures reveal Jesus to be God Himself, who is an eternal being, without beginning and without end. Lucifer is a being God created and is in the hierarchy of angels. It is also worth noting that he is a fallen angel described by scripture as a rebel against God and has been cast out of heaven.

He now resides on the earth, but with no rights here, as the earth has been given again to man through Christ's redemption, as it originally was, before the fall.

Scripture references to the morning stars singing together may refer to the sound the planetary stars may have made which creation scientists believe would have been heard on the earth up until the time of Noah's flood, when the stratosphere of earth changed and could no longer pick up these sounds.

The sons of God referred to here are angels, who scrpture explains are created beings and completely different from God Himself. They are sons because God created them.

Jesus is named by scripture as God, throughout both testaments. Lucifer is listed as envious of God's position and jealous of worship given to God, but never as God or even a God, except where the New Testament lists him as the god small g of this earth, which he took by trickery from Adam at the fall, but which Jesus took back from him at the cross and has given back to man.

The fact of Lucifer's rule of the earth is proven by the temptation he enticed Jesus with during Jesus' 40 day fast. It would not have been a temptation if Lucifer did not at that time own all the kingdoms of the earth which he told Jesus would be his, if Jesus would only worship him.

It depends on a number of factors: acceptance of the Trinity, how one defines "son", and so on. For example, there's not much room for them to be brothers in Trinitarian theology. According to the Trinity, Jesus is not a separate person from God the Father, while pretty much everyone (Trinitarian or otherwise) accepts that Satan most certainly is a separate being. Under this system, if we are to draw a family relationship at all, then Jesus would be closer to being Satan's father than to being his brother, though even that might not be appropriate an appropriate way to put things: see below.

The next question is what difference being "begotten" makes, as opposed to being "made". For Trinitarians this isn't really relevant to the original question, since we have already established that Jesus and Satan are not brothers, though one could launch into a digression over whether or not Satan should even be considered a "son" of God, and it matters to that. But for non-Trinitarians, this question carries substantially more weight, which is why I mention it here.

Some sects do not believe that there is any substantial difference between being begotten and being made: that the former is just a poetic reference to the latter. If we accept that there is no substantial difference, then Jesus and Satan are indeed brothers. Without that difference, the same reasons we call Jesus the son of God also apply to Satan. This isn't unique to the two of them, either, as they apply to all other created beings: angels, humans, and everything else. We could take comfort in the idea that Jesus is our brother, too, but there's an unsettling thought to go with the comforting one, for Satan would also be our brother. I'm reminded of the old cartoon trope, where a character's dilemma is represented by an angel and a devil sitting on his shoulders.

Other sects believe that there's a real distinction to be drawn between the two processes, and if we accept that argument, then they are not brothers. Jesus is still the son of God, but there isn't a family relationship to draw between God and anybody other than his "only-begotten son". One must then reconcile this with other references to sons of God, but different sects have their own ways of doing that. I don't know whether or not I'd find this more comforting than the no-distance argument: it distances us from Satan, but it distances us from God, and from Jesus, even more.

The Lord Jesus is the only begotten Son of the Father:

(John 1:14) And the Word was made flesh, and dwelt among us, (and we beheld his glory, the glory as of the only begotten of the Father,) full of grace and truth.

(John 1:18) No man hath seen God at any time; the only begotten Son, which is in the bosom of the Father, he hath declared him.

(John 3:16) For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life.

(John 3:18) He that believeth on him is not condemned: but he that believeth not is condemned already, because he hath not believed in the name of the only begotten Son of God.

(1John 4:9) In this was manifested the love of God toward us, because that God sent his only begotten Son into the world, that we might live through him.

Original Word: בֵּן Hebrew: Sons

As applied to Job 38:7.

Strong's Concordance Ref. word 1121. ben, bane; a son (as a builder of the family name), in the widest sense(of lit. and fig. relationship including grandson, subject, nation, quality or condition, etc.)

Son, HUIOS in Greek primarily signifies the relation of offspring to parent{W.E. Vine Expository dictionary of Bible words}, Quote: "The Lord Jesus used huios in a very significant way, as in Matt. 5:9, 'Blessed are the peacemakers, for they shall be called the sons of God,'.

[Greenman] The two words could never be confused when reading them in their original language. English does not always translate the subtlety between these two words well.

John 1:18 (NIV) indicates, "No one has ever seen God, but the one and only Son, who is himself God and is in closest relationship with the Father, has made him known."

[Greenman] Indicating there is no equal. John 1, also explains that all things were created through him. (pre-creation). Satan a.k.a Lucifer was a created being often refereed to as an angel (since fallen).

No, it would be extremely rare for anybody to think the two were brothers as that would be a teaching that would fall far beyond the pale of traditional christian teachings or doctrine.