Why don’t we see the hidden face of the moon? Is this because the moon rotation is perfectly synchronized with Earth?

You are correct. The moon is synchronized with Earth, although not perfectly. There is a wobble which I’ll show you below.

The “face on the moon” means that some people see what looks like a face on the side of the moon we can see. It’s not on the far side, but the side we see.

Here is a picture of the moon with the “face” outlined that people call the “face on the moon.” It is not meant to be comical (though it looks funny), but outlines the moon features people imagine look like a face.

More below the picture.

Now, here’s a comparison of the near and far side of the moon taken by a satellite we placed in orbit around the moon. On the left is the familiar side of the moon we see facing Earth. On the right is the side we never see from Earth because it faces away from us. This shows the moon is tidally locked with Earth so that one side always faces us. More below.

And finally, here’s an animation that shows that the moon isn’t perfectly synchronized with Earth, as you put it (scientifically it’s called “tidally locked”), but wobbles quite a bit from full moon to full moon. Due to this wobble, sometimes you can get a peak of what’s on the far side. You’ll also notice the moon gets larger and smaller in the animation. This variation happens because the moons orbit is slightly elliptical around us, so sometimes it’s a little closer and looks bigger, and sometimes a little further and looks smaller. A full moon when the moon is closest is called a supermoon.

moon-wobble

Given that NASA has landed on the Moon, why did they not put some plants on the lunar surface?

Wow. What would have happened if astronauts had brought plants to the moon? Surely by now the moon would be green! Here I always thought the thing was made of blue cheese.

You must know from the other answers that there’s no air on the moon and plants need air to survive just like you do. Another issue is there’s no water on the moon – no rivers, no lakes, no rain – and we all know if plants don’t get water they die.

However, I can offer you what none other can! Below is a picture of what the moon would look like if it really was green!

Which is more difficult to achieve 100 years from now, building a giant telescope to directly observe other planets in a distant Galaxy or building a faster than light space ship?

Faster than light speed will never be possible, ever. The best you’ll be able to do is maybe 1/10th the speed of light given an enormous amount of propellant, and then you’ll have to slow down at the end of the trip using just as much fuel as it took you to go that fast in the first place. There will be no warp drive or something that folds space. These are all convenience devices for story telling on science fiction movies.

Maybe in 200 or 300 years generational space environments will be the way to travel to other planetary systems. A generational space environment is a huge ship that rotates for a sense of gravity inside, where whole generations are born and die over hundreds of years – often resenting that their ancestors trapped them on that ship – to reach far away places.

Your given time frame of 100 years from now is not very far away in the bigger picture, but having a huge telescope on the far side of the moon is possible and would help with astronomical observations.