Christians

Especially for teenagers/young adults, it is better to avoid close friendships with people of the opposite gender whom you cannot marry. At this age, feelings develop easily, and it can lead to wrong actions or heartbreak (or both). There are also a lot of hormones that interfere with good judgment. Sometimes, being forbidden from being with someone makes you want them more and that can lead to a Romeo and Juliet type of situation which doesn't work out well for anyone. 

Also, it may lead to false hopes, e.g. fantasizing that he might convert to Islam, whereas that is unlikely. I think in general, girls tend to look at young men as potential life partners at a younger age than guys. So it is likely she might end up emotionally dependent and starting to build up the illusion that they will be together in life, whereas this is unlikely to happen in real life. At the same time, his mind might be entirely on pursuing his own goals in life and not sharing this idea. 

Islamically speaking, it isn't appropriate to be spending time alone together, going places together, being emotionally intimate with each other, touching each other in any manner, etc. 

I am not saying there is no possible benefit to the friendship as friends can help us through all sorts of life challenges, but just that it's better to be cautious and more distant.

The way to do that is just being less close and communicating less and etc. Friendships also take their own course and people often move apart in life, especially after graduation. 

Christians and Muslims come from many ethnic and racial backgrounds. The only person they are all descended from is Adam.

At some point in history, the idea became popular that Arabs are descendants of Ishmael, and Jews are descendants of Isaac. Obviously not all Arabs are really descendants of Ishmael and it seems questionable for all Jews too, so it shouldn't be interpreted as a literal fact. Anyway, most Muslims are not Arabs so this does not factor into questions of descent. 

There are texts identifying the Prophet Muhammad as a descendant of Ishmael so this may be another way of associating Islam with Ishmael. Also Muslims tend to say that Ishmael was the son of Abraham whom God asked him to sacrifice (I sense another complicated question coming...) whereas Christians and Jews say it was Isaac. So, again, one can say there is a sort of figurative association or inclination towards each.

Of course, both Isaac and Ishmael are respected in the Islamic tradition and in the Qur'an.

A scriptural perspective: According to Qur'an and hadith, it is good to share the message of Islam and invite others to it (what is known as da'wah). This should be done in a good way, with politeness and good forms of discussion and argumentation. 

However, nothing in Islamic texts that says that conversion for the sake of conversion is a goal. There is no real point to false conversion or forced conversion. The Qur'an and Islamic worldview acknowledge the existence of people of other faiths and that there is good and bad in other faiths. It is said that at the end of time when the Mahdi (the promised saviour/last imam) arrives, many people will convert to Islam because it will be clear that he has a true and good message. However, texts also indicate that some people will remain with their own religions and not convert to Islam. 

Many regions that today are majority Muslim took time to convert to Islam. This process happened gradually, over hundreds of years, and wasn't instant, even though the rulers of many of these regions identified as Muslim. Probably this is because while conversion is a social phenomenon, it is also a highly individual one, and individuals have different reasons for religious conversion. Sometimes there is a conflation between the expansion of the original Arab-Muslim Empires and conversion to Islam, but in reality just because the borders expanded did not mean that everyone in these territories suddenly became Muslim. 

A sociopolitical perspective: The Islamic world today does not have a single goal. While it would be ideal if Muslim-majority countries and Muslim leaders could unite (especially on the basics, such as economic cooperation), currently, this is not happening. Individual Muslim leaders and Muslim groups have their own goals. By and large, I think most Muslims today are more concerned about internal issues (politics, the threat of war) or questions regarding Islam and contemporary life, rather than expansion. Islam is already a large religion and isn't in danger of disappearing due to numbers. Also, most Muslim-majority countries have historically established religious minority communities. That being said, there are some Muslims who do work at spreading the message of Islam.

In any case, Islam is still spreading (particularly in the West and parts of Africa, but also - although there is less attention to it - in Muslim-majority countries and India). There is some interesting literature on the dynamics of the history of the spread of Islam if you wish to explore it.