Is celebrating Christmas required of Catholics?

While participation in the Holy Mass on Christmas is mandatory, receiving of Holy Communion is not. It is also not mandatory to have other celebrations. In fact, many members choose not to celebrate Christmas within a year of death of a family member, or someone equally dear.

No, there is no obligation for Catholics to have a Christmas tree or to give gifts. In fact, in many parts of the world, there is no gift-giving tradition for Christmas Day—for example, in many countries the exchange of gifts takes place on the Epiphany (January 6). In any event, the Church obligates no one to a strictly cultural tradition. Moreover, Christmas trees are generally only used in the Northern Hemisphere, where it is winter (in the Tropics and in the Southern Hemisphere, Christmas takes place during summer weather).

The only requirement for Catholics is to obey the precept of resting and attending Mass on Christmas, which is a holy day of obligation:

The first precept (“You shall attend Mass on Sundays and holy days of obligation and rest from servile labor”) requires the faithful to sanctify the day commemorating the Resurrection of the Lord as well as the principal liturgical feasts honoring the mysteries of the Lord [such as Christmas], the Blessed Virgin Mary, and the saints; in the first place, by participating in the Eucharistic celebration, in which the Christian community is gathered, and by resting from those works and activities which could impede such a sanctification of these days (Catechism of the Catholic Church [CCC], no. 2042; see also no. 2177, as KorvinStarmast mentions).

There is no specific obligation to receive Communion, as no. 2042 makes clear later on:

The third precept (“You shall receive the sacrament of the Eucharist at least during the Easter season”) guarantees as a minimum the reception of the Lord's Body and Blood in connection with the Paschal feasts, the origin and center of the Christian liturgy.

Previous Confession (if necessary) and receiving Communion are, of course, highly recommended on every Holy Day.

As a Catholic, you are required to celebrate the "holy-day" of Christmas, as you mentioned, by going to Mass, receiving the Eucharist, giving honor and glory to God and celebrating Jesus' birth. However, you are not required to celebrate it in the "traditional tree in the house gift giving sense". These traditions, like many traditions on Christian holidays, represent something related to the holiday, but are not a necessity. What really is important is giving honor and glory to God on that day. To sum it up, it is not required to put up a tree, decorate or give gifts, you just need to make the holiday a holy-day. If you can find other ways to celebrate that give glory to God, do them. They are much more pleasing to Him than letting what has become mostly secular occupy your mind and heart.

P.S. There's nothing wrong with the extras. As long as Jesus is the center, they make nice additions!

Christ be with you!

Christmas is a Holy Day of Obligation. As such, Catholics are expected to attend Mass on that day. (The vigil mass on Christmas Eve counts).

but are they obligated to celebrate Christmas in the traditional tree-in-the-house, gift-giving sense?

No, they are not obligated. Such celebration is optional, though most Catholics probably do that as it has become a common custom.

From the Code of Canon Law, 1247:

1247 On Sundays and other holy days of obligation, the faithful are obliged to participate in the Mass. Moreover, they are to abstain from those works and affairs which hinder the worship to be rendered to God, the joy proper to the Lord’s day, or the suitable relaxation of mind and body.

In the US, the Holy Days of Obligation are laid out by the US Conference of Catholic Bishops. (Per Code of Canon Law 1246)

January 1, the solemnity of Mary, Mother of God
Thursday of the Sixth Week of Easter, the solemnity of the Ascension
August 15, the solemnity of the Assumption of the Blessed Virgin Mary
November 1, the solemnity of All Saints
December 8, the solemnity of the Immaculate Conception
December 25, the solemnity of the Nativity of Our Lord Jesus Christ

Note that these are a subset of the fuller list of days where people are expected to attend mass, above and beyond the weekly Sunday mass. From the Catechism of the Catholic Church, in the section covering the Third Commandment ...

2177 The Sunday celebration of the Lord's Day and his Eucharist is at the heart of the Church's life. "Sunday is the day on which the paschal mystery is celebrated in light of the apostolic tradition and is to be observed as the foremost holy day of obligation in the universal Church."

"Also to be observed are
the day of the Nativity of Our Lord Jesus Christ,
the Epiphany
the Ascension of Christ
the feast of the Body and Blood of Christ,
he feast of Mary the Mother of God,
her Immaculate Conception,
her Assumption,
the feast of Saint Joseph,
the feast of the Apostles Saints Peter and Paul, and the feast of All Saints."

If you are not in the US, the Bishops in your country will spell these days out.