How has Christianity grown and developed throughout the world over time, how has it spread and how have other cultures influenced it?

It would be impossible to discuss the growth of Christianity over time without writing several books.  Throughout the years there have been people who completely accepted Christianity as true and have so cared about people in another area that they have taken the message of Christianity to them.  But the basic answer is Christianity has grown because it is not bound to one culture or people group.  (It goes without saying but any Christian will tell you that Christianity grew because it is true and though I hold that God grew the Church I feel like the question is looking for a more nuanced answer).  This is the basic problem which led to the Jerusalem Council (ca. 45 AD see Acts 15).  At this council Christians debated how much this new belief was tied to Jewish cultural heritage.  The conclusion was that Christianity was not tied to the Jewish culture and thus as long as one proclaimed the life, death, and resurrection of Jesus as the fulfillment of the promises of the God of the Bible and the need to follow this Jesus, one was proclaiming Christianity.  These are the essential beliefs and they are consistent, however, each time these beliefs encounter a new culture they are expressed slightly differently in worship.  This worship is how cultures impact Christianity, in giving us new expressions of the faith and new angles to see God.

Thus, every culture Christianity enters influences it, through how the locals express the life of Christ.  One tiny example is how Christians around the world sing, pray, or fellowship.  Each culture expresses these aspects of faith differently.  Some pray standing others sitting, some sing quietly others are boisterous, but none of this change the inherent message of Christianity.  There is a great deal of latitude for cultural expression in Christianity; consider how Jesus is depicted in local art.  There is a lot of criticism in the media that American and European Christians often represent Jesus as white, but one of the great aspects of Jesus is he takes on all cultures.  My wife and I collect nativities from around the world, each one depicts Jesus in that cultural setting, and is a wonderful reminder that Jesus belongs to all people precisely because he transcends all people.  What Christianity seeks is a mosaic of cultures, which all represent the God of the Bible take the basic message of love God and love creation into the world, and produce a kaleidoscope of expressions to show the majesty of God.

For more on this see

Justo Gonzalez The Story of Christianity vol 1 & 2

Ruth A. Tucker From Jerusalem to Iranian Jaya

Bruce Shelly Church History in Plain Language

Robert Weber Ancient Future Worship

Rory Noland The Heat of the Artist

The Oxford History of Christian Worship