While welcome here, this might reach more informed audiences at r/askanthropology and r/linguistics

You can speak an ancient people's language without being their descendant

PIE descendants are spoken in most of Europe, India and Iran. When PIE first started developing, there would have been many other languages throughout these areas, so why did PIE spread across them, as opposed to another language.

This thread relates two of the leading theories on Indo-European dominance in Europe:

http://www.reddit.com/r/AskHistorians/comments/1s25dp/would_the_inhabitants_of_europe_before_the/

tl;dr: The Indo-Europeans either brought farming to Europe (and possibly to South Asia) or they brought pastorial nomadism based on horses. Movement is easier (especially on horseback) through the steppes where the PIE speakers are usually said to originate, so they moved around the world very quickly. Pre-Indo-European groups wanted access to their reliable way of providing food for themselves and the easiest way to do that is to assimilate the language and culture.

Alternatively, why does one language root dominate so much of the world; why isn't there lots of language families in these areas?

As /u/grapp pointed out, you don't have to be genetically linked to a group in order to speak their language or take on their culture.

Now once Indo-European culture had taken root in the places we know today, we have to move on to other reasons for their dominance today. The only source I can cite is Guns, Germs, and Steel, which most historians (at least on this sub) consider pretty good pop history but riddled with too much conjecture and not as rigorous as they would like.

But I think that book is a good introduction to some basic hypotheses as to how Europeans came to dominate the world. I'll summarize a few points the book makes.

Food crops can move laterally relatively easy because of the way the sun hits the Earth. Temperature stays relatively the same laterally, which allows the crops from the classical river valley civilizations, the Mesopotamians and Egyptians, to move to Mediterranean Europe or to South Asia. So we can establish that Indo-Europeans managed to get a great spot for the future development of agriculture. This particularly helps explain the population boom on the Indian subcontinent and while they may not have exported their culture and dominated two whole continents like the Europeans did, they still constitute 1.2 billion people who have an Indo-European culture and close to 1 billion of them speak a IE language (I subtracted the 200 million Dravidian speakers from India's population).

They were also more exposed to ideas from other cultures, with Persian and Indians banking on trade along the Indian Ocean, while the Europeans had the Mediterranean. Compare this to China, who were in relative geographical isolation with the Gobi Desert to the East, Himalayas to the South, and frigid steppes to the north. They also lacked trade partners directly east in the Pacific Ocean with Japan, Korea, and maybe a few Malayo-Polynesians being the only major ones. The Chinese had a "head start" on agriculture thanks to rice and soy, two high yield and nutritious crops, but due to their geographical isolation they developed a cultural attitude of isolation.

When your culture is exposed to more and more cultures, you tend to want to create links with them. Europeans could get what they could not grow or create in their own part of the world through trade. Europeans as a whole developed a cultural attitude inclined towards exploration and finding those trade links (and eventually conquering those trade links). Combine this with favorable ocean currents that can take them from the Old World to the New World more easily than anybody on the West African coast and the rest is disease and colonial expansion.

As a follow up question, would English, Chinese and Nahuatl all be related, distantly? Are all human languages descended from one common ancestor, or did language emerge independently in multiple locations?

Sadly we will never know the answer to this. Humans have been migrating out of Africa as early as 130,000 BC, and written language was first developed roughly 3200 BC.