The differences between people in the same country are bigger than the differences in people between countries. Scientists can talk about averages, saying that people in country X are taller, on average, than people in country Y. But people have many different heights in both countries. Maybe the tallest person in country Y is taller than the tallest person in country X.

About the closest that science can get is in comparing skulls. Forensic anthropologists can often tell if a person had mostly African, European, or Asian/Native American ancestry by looking at several measurements on a skull. They are never 100% sure, though. Here is a site from the Smithsonian Institute that can allow you to try. It is based on finding skulls in the Chesapeake Bay area of what is now Virginia. The skulls were from the 1600’s or early 1700’s.

People in the same country may have ancestors from different countries. The borders of countries change over time. People move from place to place. Most people have ancestors from multiple places, whether they know it or not. This makes it even more complicated to determine a person’s ancestry from physical characteristics.

If you’re interested in questions like this, you may enjoy this web comic that the Smithsonian Institute created:
webcomic.

That's a great question. Your genetic material or DNA (deoxyribonucleic acid) is responsible for much of what you look like. Your DNA is composed of genes which are specific areas of the DNA that code for specific proteins (and sometimes other molecules that play an active role). These proteins are what cause these differences to occur. Different genes control all of those different traits that you talked about. These characteristics can be very different between people from different countries. This is because there is a tendency for people, more so in the past (like think hundreds of years ago) to remain in the area that they were born. And so you tend to have the same DNA that is shared within the same area. Also if we go back even further like thousands of years ago, these particular genes were selected for due to the conditions of the environment.

So for example, in areas near the equator (where there is a lot of sun), genes that cause darker skin tend to be more common. This is because the darker skin helps to protect against the sun's UV rays that can cause a lot of damage and result in diseases like cancer. Different eye shapes lend to better protection against extreme cold/wind or heat and so that's why some people from different countries have different eye shapes.