In medieval Byzantium as in the West, the Christmas season was another period of carnival, or the raucous celebration of transgressive humor to release the tension of strict hierarchy and rules the rest of the year (tl;dr - Mardi Gras).

A note of caution is necessary when dealing with contemporary descriptions of Kalends, the feast of fools, Christmas carnival, etc--basically every report is going to be from a cleric complaining about the festivities. Nevertheless, a few patterns emerge beneath the hostility.

First, clergy and laity participated in the celebrations. In fact, churches themselves were often a stage for some of the events! Theodorus Balsamon is a bit later than you're asking about, the second half of the 12th century rather than the first, but here's a bit of his rather cranky description:

Disguising themselves for various roles, certain clerics step into the center of the church [Balsamon is most concerned with the Hagia Sophia], wearing swords and dressed as soldiers. They go on dressed as monks or four-legged animals. The superintendents snap like charioteers, or paint their faces and mimic women, or do other shameful things to provoke spectators to laughter. The unlearned are moved to laughter by pouring wine into pitchers [n.b. maybe this is a play on the Eucharist?] and are allowed to chant Kyrie eleison in response to ludicrous verses. ["Lord have mercy", a call-and-response in Christian liturgies].

Balsamon's target here is explicitly the involvement of the clergy to such an extreme degree. In Sacred Folly: A New History of the Feast of Fools (which the January Kalends/Christmas carnival is not), Max Harris argues that this much, public, and prominent clerical involvement was new to the 12th century, although there are some traces of it peeking up through the sources by the late 11th century.

However, there is a LONG and rich tradition of Greek theologians and Church councils excorciating lay participating in Christmastide carnival-type celebrations, which they viewed as hearkening too dangerously back to pagan rites. (#WarOnChristmas #KeepTheXInXmas) The practice of good Christian men dressing up as animals and as women is of particular concern. From sources before and after the 12th century, we have the general idea that carnival involved organized or unorganized processions through the streets, shouting and singing and dancing.

One imagines, of course, that there was also some religion going on. ;)

OK, I'm Greek, so I'll tell you what I know from history classes, religion classes and from tradition. I don't know about the specific era, but I assume not much changed.

The Christian Romans didn't celebrate Christmas on the 25th, up until the 5th Century. Instead they celebrated around 7 of January (some sources say 5 of January), along with the 6 of January celebration of the Epiphania (ascension of Virgin Mary to heaven; I don't know what other denominations believe, Orthodox believe that she didn't die and instead she just ascended to heaven). The 25th of December was the day when pagans celebrated Apollo/The Sun.

According to a bishop named Saint Chrysostomos, in a speech he gave at Antiocheia in 386, around that time the Christmas celebration was moved on 25th of December to coincide with the pagan celebrations, presumably to force Christians who also followed some pagan beliefs to stick to Christianity. We also have a speech by another bishop, Saint Gregorios that he gave on 25th of December 380 where he says "Today Jesus is born, praise him". These two were the greatest supporters of making Christmas celebrations take place in 25th of December. Of course the move was met with dismay and some revolts from non-Christian citizens.

As a result, some parts of the Sun celebration were carried over to the Christmas celebration. Of course everyone would go to church, and it was a very popular day to receive communion. It was made traditional to fast 40 days before the celebration so you could receive this communion.

As part of the church ceremony, the faithful would build a manger with straw and place fake animals around an icon of Mary holding baby Jesus. Someone would hold a candle over the manger to symbolize the Star of Bethlehem.

The Emperor would appear in public with a light halo around his head (originally part of Sun celebration, the halo symbolizing the God of the Sun, now symbolizing Jesus). He would exit the palace and head over to the church (after its construction, Agia Sofia became the sole church the Emperor would go to). He would be accompanied by other nobles, servants and bodyguards who would lay red cloths and palm leaves on the ground for him to pass. The people would gather to watch the procession and would sing specific hymns to him, to Jesus and to God. When the Emperor arrived at the church, the Diakon (junior priest, responsible for candles, processions and performing some rituals) would remove the halo from the Emperor's head and he would proceed to meet the Patriarch. Then together they'd head over to the altar and the Emperor would take his place on the moutatorion (an elevated seat reserved for the Emperor), where he would sit until Communion. He would be the first to receive Communion. On his return to the palace, Prasinoi and Venetoi (political "parties") would also sing him praises at the church's steps and in 5 other different parts of the city. By the way, these hymns were both in Greek and Latin initially, later giving way more to Greek, but some Latin retained.

Children would go around the neighborhood with an orange or an apple and knock on doors. The people opening would shove a coin into the fruit as a gift (Sun celebration). This was later attached to the 6th of January celebration of Epiphania, and instead of fruit, the kids would just say the phrase "Goodmorning, Good Lights and the Pulustrina first" (Good Lights = euphemism for Epiphania; Pulustrina = gift). We still do this today.

This was copied for the 25th of December celebration, but now kids would go around houses, starting 12 days before the celebration date, usually in the afternoon or early night and would sing the Kalanta, a song about Jesus's birth. At the end of their performance the household was supposed to give them either a coin or some sweets or both, usually according to the social status of the family (i.e. a rich family was supposed to pay coin, a poor family was acceptable to pay with sweets). The singers wouldn't leave unless they received payment first. They would also improvise compliments to the house lord or lady.

By order of the province governor or the Emperor for Constantinople, the streets were cleaned and decorated. Usually the decorations included leaves of rosemary, myrtle and seasonal flowers. Some people liked to dress up in masquerade and go around celebrating, especially at night. In Constantinople, these masqueraded celebrators would try to be as annoying as possible to passersby and neighbors, in an effort to drag them into their celebrations as well.

When the Emperor returned to the palace after the church ceremony, a large banquet would be organized where every noble, general and palace servant was invited to, as well as all foreign dignitaries. It was also traditional to invite 12 poor people, in honor of the 12 disciples of Jesus.

After this banquet, if the Emperor gave permission, race horses would be held at the Hippodrome and the Emperor and all his guests would attend. The people were very excited about these races and especially on this day where the prizes were at their highest and the honor to the victor was the greatest.

Another banquet would be held after the races and other nightly celebrations according to the Emperor's instructions.

The common people would spend the days leading to Christmas making sweets and special foods reserved only for that day. The basis of all of them was usually honey. Particularly interesting was a special kind of soup made for pregnant women that was supposed to strengthen the baby and the mother's milk. The church tried to ban this tradition because they considered it blasphemy, as it was done in Mary's honor and as the dogma went, Mary wasn't pregnant in a regular way. Some of the sweets we still do today.

Notice that beyond some coin or sweets, presents weren't around for the celebrations.

Initially New Year's for the Romans was 1st of March. Later on it was made the 1st of January, so it would be closer to Christmas (the week after). On January 1st is Saint Vasileios' day. Saint Vasileios was a priest in Syria, and as the legend holds, bandits (or Saracens) tried attacking his city. The city tried to negotiate with them, and the bandits asked for all the gold in the city, in exchange for leaving them alone. Saint Vasileios organized gathering all the gold coins, jewelry and other items and gave them to the bandits. However, the bandits saw a vision of Mary or Jesus or Angels (depending on the version) and left the city leaving all the loot behind. Saint Vasileios didn't know who gave what exactly so he could return the items. So he had the city bakers make a pie for each household, and in each pie he stuck some of the gold items. He then gave one pie to each household. Miraculously everyone received exactly what they gave, and Vasileios was declared a Saint after this.

So, during New Year's we celebrate Saint Vasileios, and the Byzantines started a lot of traditions around this. First and foremost is making a pie that is only done during that day. Each household is supposed to make one. The master of the house is responsible for cutting the pie symmetrically, and to designate the first piece for Jesus, one subsequent piece for each absent family member, one piece for dead family and then the next pieces go in hierarchical order to the present family, starting from himself, going to his wife and then the children in order of birth and then to any guests present. Inside the pie is a gold coin, and whoever takes the piece with the coin is supposed to have good fortune for the rest of the year. After eating the pie, each attendant is supposed to give a gift to the rest in spirit of Saint Vasileios' selfless spirit.

A carry over from the Sun's celebrations that got into the New Year's eve celebrations is throwing olive leaves on charcoals and making a wish or prayer. If the olive leaf crackles and jumps, then it's a good sign that your wish is granted or your prayer was heard. If the leaf withers and burns away with a hiss, then it's a bad sign.

There was widespread celebration in the streets and in each household on both days, and usually the feasts included lamb meat, as the lamb symbolizes Jesus. The Byzantine Romans believed that the way you spent New Year's Eve was the way you would end up spending the rest of the year. So they'd celebrate as much as they could so they could have a happy year (at least this was the pretext). As with Christmas day, on New Year's eve and New Year's they'd masquerade and go around singing, drinking and playing games as well as annoying anyone who wasn't joining the celebrations. This was of course met with dismay from the Church, who viewed it as too pagan and inappropriate for celebrating Jesus' birth.

On both days the Emperor would decree that no one should be arrested for small and petty crimes, but the law still applied for serious crimes. In the spirit of the days, all prisoners who were serving sentences over such small crimes, or prisoners who had very little time left to serve were released.