Will an alien invasion by secular and militaristic extraterrestrials solve a lot of problems on Earth?

An alien invasion by secular and militaristic extraterrestrials would solve one of the biggest problems we have here on Earth. That problem being: does life exist on other planets? Finally we would have an answer!

Shouldn't we be looking for life on planets that formed over 10 billion years ago as they've had more time for life to form?

Planets that formed over 10 billion years ago are about 5 billion light years from us, meaning that if we could somehow study planets that far away (we can’t) the light reaching us from there would have taken 5 billion years to reach us. Therefore, we’d be seeing what the planet looked like 5 billion years ago, not what it looks like now. Below is what we are able to see of the most distant galaxy that we have seen to date. It’s about 13 billion light years and we are just seeing the early stages of development because it took light 13 billion years to reach us.

As you can see from the image, Hubble can barely make out the galaxy, what to speak of any planets that might be orbiting the billions of stars inside that galaxy. The best we can do is study planets in our vicinity of the Milky Way.

What's the use of the universe if Earth is the only habitable planet?

First, we don’t know Earth is the only habitable planet. However, until we have some proof, we have no way of knowing for sure. That’s why we’re searching so hard for extraterrestrial organisms.
Second, who says the universe has to have a use? It just is. Similarly, you could say what’s the use of living if you die in the end. That kind of view doesn’t do anyone any good. Just go with it and enjoy the exploration and discovery life offers us, like whether or not life exists elsewhere in the Universe.

How can we believe in God, but not in alien/extraterrestrial life or vice versa? This seems like a paradox to me.

I don’t see either as being dependent on the other. In other words, you can believe in God and not in extraterrestrial life. Many religious people do in fact hold that view. On the other hand, you can believe in extraterrestrial life and not believe in God. I know many people who hold that view.

So I don’t see a paradox at all. According to the Google dictionary, a paradox is “a seemingly absurd or self-contradictory statement or proposition that when investigated or explained may prove to be well founded or true.” I fail to see a contradiction that two different people can hold either view. It’s all theory.

In reality, we do not know for sure that life exists anywhere other than Earth because we have never found any evidence of life, although we continue the search. We have yet to find even a fossil of a microbe from space. However, the laws of mathematical probability suggest that with billions of planets in this galaxy and billions of other galaxies, it is likely that there is at least microbial life somewhere, if not something more advanced. We just haven’t the proof yet.

Similarly, except for subjective experiences, there is no proof that God exists either, although people believe that God exists. Maybe both exist. Maybe neither exists. I suppose it’s a question of faith in either view.

Using our current technology, how close to us would an alien spacecraft have to be before it could detect Life on Earth?

During Earth’s night you can see the lights of major cities from the moon. So this would imply that further out from the moon one could detect light. Then there are the radio and television waves. They go out infinitely into space, but they become four times weaker every time the distance doubles. In any case, we pick up radio signals from Voyager I and II even though they have left the solar system, so I would suppose people could pick up our radio signals quite far away.

If aliens in our galaxy are exponentially super advanced, why have they not completely colonized the entire Milky Way by now?

Well, let’s not put the cart before the horse. Right now we don’t know for sure that any life exists anywhere other than Earth. We suppose it does by mathematical calculations, but we haven’t even found a fossil of a microbe from space.

Secondly, suppose there is some super advanced civilization (your use of “exponentially” doesn’t seem to make any sense), the reason they probably haven’t completely colonized the entire Milky Way Galaxy is that for them, as for us, the laws of physics make it pretty much impossible to achieve interstellar travel because of the vast distances involved. And even if they could colonize some surrounding systems by now, it would take light and radio waves 100,000 years to reach us from the other side of the galaxy.