How widely held among Protestants is the belief that the authors of the books of James and Jude were half brothers of Jesus?

Many Protestant churches I've visited, myself included, believe that Jesus had brothers, four of them - named James, Joseph, Simon, and Judas - all of which we believe to be the authors of the Book of James and the Book of Jude. Some Protestants believe them to be children from Joseph's prior marriages (which I don't believe I've ever seen Scripture supporting), or being born after the birth of Christ.

Why? Let's look in to Scripture. First we start with Matthew 1:25, which indicates it very possible (and as Jews, very likely) that Joseph and Mary had other children:

"But he [Joseph] knew her [Mary] not until she had given birth to a son. And he called his name Jesus."

Those familiar with the language at that time will tell you "knowing" a person, in this context, is referring to having intercourse. As Jews, and followers of the Law (which these same Protestants believe Mary and Joseph had to be in order to be so blessed), they would have been taught the Commandments given to Adam and Noah along with the Commandments laid out in Moses' books - the first two (Adam and Noah) containing the Commandment to "be fruitful and multiply".

We also see Jesus' brothers in John 7:1-10

After these things Jesus walked in Galilee: for he would not walk in Jewry, because the Jews sought to kill him. Now the Jew's feast of tabernacles was at hand. His brothers therefore said unto him, Depart hence, and go into Judaea, that thy disciples also may see the works that thou doest. For there is no man that doeth any thing in secret, and he himself seeketh to be known openly. If thou do these things, shew thyself to the world. For neither did his brothers believe in him. Then Jesus said unto them, My time is not yet come: but your time is always ready. The world cannot hate you; but me it hateth, because I testify of it, that the works thereof are evil. Go ye up unto this feast: I go not up yet unto this feast: for my time is not yet full come. When he had said these words unto them, he abode still in Galilee. But when his brothers were gone up, then went he also up unto the feast, not openly, but as it were in secret.

And again, this time with Mary, in Acts 1:14:

These all continued with one accord in prayer and supplication, with the women, and Mary the mother of Jesus, and with his brothers.

This article sums up how our belief is different from that of Roman Catholics, and also references the verses you provided as well as the ones I cited in this answer:

Some Roman Catholics claim that these “brothers” were actually Jesus’ cousins. However, in each instance, the specific Greek word for “brother” is used. While the word can refer to other relatives, its normal and literal meaning is a physical brother. There was a Greek word for “cousin,” and it was not used. Further, if they were Jesus’ cousins, why would they so often be described as being with Mary, Jesus’ mother? There is nothing in the context of His mother and brothers coming to see Him that even hints that they were anyone other than His literal, blood-related, half-brothers.

A second Roman Catholic argument is that Jesus’ brothers and sisters were the children of Joseph from a previous marriage. An entire theory of Joseph's being significantly older than Mary, having been previously married, having multiple children, and then being widowed before marrying Mary is invented without any biblical basis. The problem with this is that the Bible does not even hint that Joseph was married or had children before he married Mary. If Joseph had at least six children before he married Mary, why are they not mentioned in Joseph and Mary’s trip to Bethlehem (Luke 2:4-7) or their trip to Egypt (Matthew 2:13-15) or their trip back to Nazareth (Matthew 2:20-23)?