Do Muslims celebrate Christmas like Christians?
No. Muslims do not celebrate Christmas like Christians. In fact, we do not know why Christians celebrate it when it has a pagan origin.
“No such festival as christmas was ever heard of until the THIRD
century, and not until the FOURTH century was far advanced did it gain
“Long before the fourth century, and long before the christian era
itself, a festival was celebrated among the HEATHEN, at that precise
time of the year, in honor oft the birth of the son of the Babylonian
queen of heaven; and it may fairly be presumed that, in order to
conciliate the heathen, and to swell the number of the nominal
adherents of christianity, the same festival was adopted by the Roman
church, giving it only the name of Christ. This tendency on the part
of christians to meet paganism half-way was very early developed.”
Source: Hislop, A., The Two Babylons, Loiseaux Brothers, Neptune, N.J. p. 93
I believe it is more of a custom thing than religious.
What do they do on Christmas?
It's just like any other day for Muslims.
Does Islam reject Jesus?
No, not at all. All Muslims believe in Jesus or Eesaa, peace be upon him like they believe in all prophets and messengers sent to humanity. However, we reject what the various factions of Christians say about him i.e. he is the son of Allah or he is Allah or he is the third of three.
Allah (al-Ilaah, Eloh-im), the Sublime and Exalted, said in his final revelation to mankind, establishing the truth about Jesus, the Messenger (peace be upon him):
يَا أَهْلَ الْكِتَابِ لاَ تَغْلُواْ فِي دِينِكُمْ وَلاَ تَقُولُواْ
عَلَى اللّهِ إِلاَّ الْحَقِّ إِنَّمَا الْمَسِيحُ عِيسَى ابْنُ مَرْيَمَ
رَسُولُ اللّهِ وَكَلِمَتُهُ أَلْقَاهَا إِلَى مَرْيَمَ وَرُوحٌ مِّنْهُ
فَآمِنُواْ بِاللّهِ وَرُسُلِهِ وَلاَ تَقُولُواْ ثَلاَثَةٌ انتَهُواْ
خَيْرًا لَّكُمْ إِنَّمَا اللّهُ إِلَـهٌ وَاحِدٌ سُبْحَانَهُ أَن
يَكُونَ لَهُ وَلَدٌ لَّهُ مَا فِي السَّمَاوَات وَمَا فِي الأَرْضِ
وَكَفَى بِاللّهِ وَكِيلاً
O people of the Scripture (Jews and Christians)! Do not exceed the
limits in your religion, nor say of Allaah aught but the truth. The
Messiah Eesaa (Jesus), son of Maryam (Mary), was (no more than) a
Messenger of Allaah and His Word (through which He created him), (a
Word) which He bestowed on Maryam (Mary) and a spirit (Rooh) created
by Him. So believe in Allaah and His Messengers. Say not: "Three
(trinity)!" Cease! (it is) better for you. For Allaah is (the only)
God (worthy of worship), Glory be to Him (Far Exalted is He) above
having a son. To Him belongs all that is in the heavens and all that
is in the earth. And Allah is All-Sufficient as a Disposer of affairs.
Do Muslims believe he was a prophet? Or a leader, or any sort of deity?
Muslims believe, what Qur'an says about him, i.e. Jesus Is the Prophet and Messenger who called people towards Monotheism like other Prophets and he is the Messiah, the Word and the Spirit of Allah, the Exalted.