My assumption is that there would be some type of shelter for the sheep. It wouldn't be a good practice for someone who wanted to raise sheep not to shelter them during inclement weather. Stables have been around for quite a while.
As far as lambing goes, the sheepherder can control when it occurs. But if they lamb in winter they better have the lambs in shelters because they are much more susceptible to weather extremes.
Scripture itself tells us about the winter time in Judea.
[Ezra 10:9-13] 9 Within the three days, all the men of Judah and Benjamin had gathered in Jerusalem. And on the twentieth day of the ninth month, all the people were sitting in the square before the house of God, greatly distressed by the occasion and because of the rain. 10 Then Ezra the priest stood up and said to them, "You have been unfaithful; you have married foreign women, adding to Israel's guilt. 11 Now make confession to the LORD, the God of your fathers, and do his will. Separate yourselves from the peoples around you and from your foreign wives. 12 The whole assembly responded with a loud voice: "You are right! We must do as you say. 13 But there are many people here and it is the rainy season; so we cannot stand outside. Besides, this matter cannot be taken care of in a day or two, because we have sinned greatly in this thing.
The "ninth" Hebrew month is "Kislev" occurring in November/December on the Gregorian calendar and is indeed the beginning of the rainy season.
[Song of Solomon "Songs" 2:11] 11 See! The winter is past; the rains are over and gone.
Livestock to this day are not left out to pasture much past October in the Holy Land. The time of the birth, more realistically, would be during the Fall Festivals of The Lord.
Thank you for you post, Douglas.