What challenges do you face as a global project manager?

Yasmina -

Aaron's covered a lot of my thoughts, but one which touches all three of your questions relates to time and how it is perceived in different areas of the world. This is especially challenging when leading a project where you have team members in different countries with radically different perceptions of time and timeliness.

Kiron

As Roland hinted at, culture is very much embedded in languages. Idioms, body language and personal distances vary widely and can quickly cause problems.

I have been an international program manager for a good many projects, sometimes remote, sometimes in situ. (France, Germany, UK, Malaysia, Korea, Japan) those were all multimillion dollar projects. My conclusions:
For small projects, it is difficult and expensive to work in situ, and remote is normally the only option. However, I think language is the main issue (see below) , and if the PM does not speak the language someone who does should always be delegated to interpret. ( in-situ AND remote)
I am fortunate that I speak a bunch of languages. But Korean was not one of them, and I was PM in-situ. If you are not in command of the language, you are nit in control and you need to request others to speak in English. However I found that, even so, people invariably revert back to their mother tongue: you are lost. I hired a multi-language local person, but because our projects were quite sophisticated, many a time that person did not recognize the importance of discussion items. It was so important that I went to Korean language courses. ( six months). The results were spectacular: not only was there a high regard for this by customers and suppliers alike. I also did not tell my counterparts that I had gone to school, and found out what people were talking talked about behind my back during meetings without my interpreter.
Later on as a CEO, for a project in France, I insisted that the relevant marketing people as well as the subsequent program manager, spoke French. ( for the PM we had to hire from outside)... even though our counterparts in France spoke English.

And it is difficult to understand the culture if you do not understand the language. sorry for the long answer

1) One of the biggest challenges can be understanding the social cues. Try navigating office politics when you don't understand the culture or the language.

2) It didn't take long to realize that most of my time would be spent either in the hotel or office. I was fortunate to travel with coworkers who were experienced in traveling to our destinations, but in some ways it was also limiting - I went with their routines instead of exploring on my own. If I had been a little more adventurous, I'd have more to talk about than restaurants. For example, while in Korea, I was staying in a hotel in Gangnam district during World Cup. I heard music through the window, but didn't see anything when I looked out. If I had just gone outside and literally around the corner, I would have found a street party where Psy was performing.

3) It wasn't required, but would have been helpful if I had known the language before traveling to other countries. It would have been nice to be able to communicate more effectively, and fun to see people's response when they find out the giant American understands them. While a friend/coworker was in Vietnam, he was looking for some clothes in a street market and the vendor said something about not having King Kong sized clothes. That would have been fun to witness. It's also the simple things, like being able to ask that the McDonalds in Tokyo NOT put ketchup on my sausage egg mcmuffin, or knowing they were going to, in the first place.

Aaron and Kiron made valid points.
I would say culture really matters.