Christmas is a Pagan holiday... just like Easter.
While I don't deny Christmas has pagan roots, most of what is in those quotes is false.
I know of no justification for identifying those mortal biblical figures with pagan gods.
The name Easter (in English, but not any Romance languages) is pagan in origin, and popular celebrations of it include pagan elements, but neither the celebration of the resurrection nor its timing is pagan. It comes from the crucifixion happening around Passover.
Although the Germanic goddess Ēostre (for whom Easter is named) and Semitic goddess Ishtar may have both been associated with spring and fertility, I don't think they are really connected.
I don't think any of those sun gods were actually believed to be born on December 25. Those that were born at all were typically born in the spring, often in a cycle where they die every autumn and are reborn every spring. December 25 became an important date in the Roman worship of Mithras (or rather Sol Invictus, as I believe this came after he was merged with several other sun gods; Mithraic cults in the east knew nothing of this holiday), but it was not his birthday. If I recall he was born in August, and by born I meant he came forth fully formed from a solid rock without any parents. The December 25th celebration is likely a practice they borrowed from the cult of Bacchus, and may well have happened
after Christians started celebrating Christmas on that date.
I believe the "Queen of Heaven" actually refers to Asherah, the wife of El, or Baal, or YHVH himself in some rather blasphemous folk beliefs of the Israelites. (Many modern scholars and archeologists believe that YHVH was once considered a male chief deity with this female consort and children like in other middle eastern pantheons. There is evidence that many in Ancient Israel believed this way too, but that was not their original belief a form of idolatry especially offensive to the Lord.)
While the Christmas tree likely does have a (Northern European) pagan origin, the trees decorated with silver and gold mentioned in the bible were not Christmas Trees but Canaanite/Philistine idols. While neighboring peoples idols were usually made of stone or solid metal, the natives of Canaan used wooden carvings that were often covered in gold or silver leaf. The "worked with an axe by the hands of a craftsman" part indicates that the trees are not just felled, but carved into images of false gods. The very next verse refers to them as idols that are like scarecrows in a cucumber field. I see no good reason why we should have a Christmas tree, but this verse is not a condemnation of the practice.
It is not really clear when Christ was born. Likely dates range from March to October. Practically the only thing that is clear from the bible is that it was probably not in the winter.
It is also clear that many early Christians considered birthday celebrations in general to be pagan and so did not care when Christ's birthday was.
