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Muslims have certainly lived in Christian lands as well.
Islamic scholar Bernard Lewis disagrees with you. That is something which only happened in very recent times. Lewis says in his book, "What Went Wrong?" that Muslims hated coming to the backwaters of Europe, and that in fact until the decline of the Ottomans, Jews much preferred the tolerance of Islam to the murderous antagonism of Christendom.

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Universities, science, humanitarian aid, hospitals, and democracy are all innovations that came from the Christian community.
Universities, democracy, and science came from ancient Greece, a society which both Christendom and Islam claim as their root.

The word "hospital" comes from the Knights Hospitaller, a sect of Christians who helped the injured on the battlefields of the Crusades. But for most of the past 1000 years Muslim doctors were much superior to Western doctors.

"Humanitarian aid" is a modern concept hardly unique to Christian countries.

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Christianity is far more peaceful at its root, and in its first 300 years, never had an army fight and conquer in its name.
You're being very selective with your dates. Christians were in sufficient numbers to gather an army in their first 300 years - the Romans would have annihilated them if they had.

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And the Inquisition and Crusades were also a direct result of Islamic invasion deep into European/Christian territory. It was more of a taking back what had been taken from them, with some obvious excesses that were done in the name of Christianity, but were clearly more political than religious.
What nonsense! The Spanish Inquisition was established as a pogrom against Jews and the few remaining Muslims in Granada - see http://www.wikipedia.org/w/wiki.phtml?search=Spanish+inquisition&go=Go

As for the Crusades, from www.wikipedia.org

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The Crusades were a series of several military campaigns sanctioned by the Pope that took place during the 11th through 13th centuries. They began as Catholic endeavors to capture Jerusalem from the Muslims but developed into territorial wars.


Historical background
The initial conquest of Palestine by the forces of Islam did not interfere much with pilgrimage to Christian holy sites such as Jerusalem, Bethlehem, and Nazareth. However, in the year 1004 the Fatimid caliph of Cairo, Hakim, had the Church of the Holy Sepulchre destroyed. His successor permitted the Byzantine Empire to rebuild it, and pilgrimage was permitted again.

The decisive loss of the Byzantine army to the Seljuk Turks at the Battle of Manzikert in 1071 brought the beginning of Byzantine pleas for troops and support from the West.


Reputation and evaluation
In Western Europe the Crusades have traditionally been regarded by laypeople as heroic defensive enterprises, although not all historians have agreed. In the Islamic world, however, the Crusades are regarded to this day as cruel and savage onslaughts by Christendom on Islam, and so, for example, some of the rhetoric from Islamic fundamentalists uses the term "crusade" in this emotional context to refer to Western moves against them. Eastern Orthodox Christians also see the Crusades as attacks by the West, especially because of the sack of Constantinople by the Fourth Crusade.

There is an interesting symmetry between the terms "Crusade" and "Jihad". In the West the term "Crusade" has positive connotations (for example a politician might use rhetoric such as "a crusade against illegal drugs") while the term "Jihad" has negative connotations associated with fanatical holy war. In the Islamic world the term "Jihad" has positive connotations that include a much broader meaning of general personal and spiritual struggle, while the term "Crusade" has the negative connotations described above. Thus to correctly translate nuances of meaning, the use of "Jihad" in Arabic should be translated to "Crusade" in English while use of the Arabic term for "Crusade" should be translated to "Jihad" in English.

In truth much of what the crusaders did was less than heroic. They committed atrocities not just against Muslims but also against Jews and Christians. For example the Fourth Crusade never made it to Palestine, but instead sacked Constantinople, the capital of the Christian Byzantine Empire. This crusade served to deepen the already hard feelings between Eastern Orthodox Christianity and Western Christianity. The Byzantine Empire eventually recovered Constantinople, but its strength never fully recovered, and the Byzantine Empire finally fell to the Ottomans in 1453.
And then, you contradict yourself in two adjacent sentences:

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Islam, again, was violent in its spread from the time it began.

There have been peaceful and violent periods in both Muslim and Christian history.
And finally, we see your prejudices and ignorance laid out in full:

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But at its root beliefs, Islam advocates violence in the name of Islam. Christianity does not. They are NOT comparable religions.
But see this:

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Islam assigns Jews and Christians (and certain other, smaller, religions) the status of people of the Book on the basis of their monotheism, and their beliefs about God and the world. This status is based on several passages from the Quran that say how Christians, Jews, and Muslims share common scripture, morals, and prophets. Muslims believe that the 'People of the Book,' if they are decent and good, regardless of the fact that they are not Muslim, will go to Heaven. They are seen as cousins in the family of believers, and Muslims are encouraged to live on peaceful and equitable terms with them.

In one verse of the Quran, it says "God forbids you not, with regards to those who fight you not for [your] faith nor drive you out of your homes, from dealing kindly and justly with them; for God loveth those who are just." (Quran, 60:8) which is interpreted as a clear admonition not to be disrespectful or unkind to non-Muslims. According to an authentic hadith, Muhammad said to his people "The one who murders a dhimmi(non-Muslim under protection of the state) will not smell the fragrance of Paradise, even if its smell was forty years travelling distance" [Sahih Ahmed].