You are correct- the moon is moving away, and the sun is shrinking. Both changes are small, and though I don't know the exact numbers, they should both be no more than a few centimeters per year. To put that in perspective, the moon is 40 billion centimeters away and the sun is 140 billion centimeters wide.

The moon was indeed touching the earth about 4 billion years ago. In fact, Earth and Moon were formed from a collision of a Mars-size planet (proto-Moon) and a Venus-size planet (Proto-Earth). In a head-on impact proto-Moon would have been destroyed, but this impact was only a glancing blow. Proto-Moon lost most of its core and a lot of its rock was skinned off; these fell onto Proto-Earth and increased its size. The heat of impact completely melted both worlds.

The earth had a thick atmosphere of BOILED rock, and it glowed like a star for millions of years. If there was any life before the collision, certainly it would have all been fried. This amazing scenario was deduced from studies of moon rocks and is supported by computer simulations of the collision.

After the collision the moon orbited close to Earth. The moon's gravity pulled a bit more strongly on the near side of the earth. This effect caused strong tides (dragging big lumps of ocean around the world), and it also made the Earth slightly egg-shaped. As the nonspherical Earth rotated, its uneven gravity caused the moon to gradually spiral outward.
Energy was transferred from the Earth's rotation to the Moon's orbit.
Even now, our days are getting longer by 38 billionths of a second per day. This is due to the Moon's gravity giving the Earth an uneven shape, which causes uneven gravitational interactions, which drain Earth's spin and fling the Moon outward. This effect was much stronger when the moon was closer: When the Moon was at a tenth of its current distance, there were mile-high tides and the days were only a few hours long.

The sun didn't burn us in the distant past because despite being larger, its surface wasn't as hot. In fact, the Earth used to be colder, because the sun put out 30% less heat. In a few billion years the sun will run out of hydrogen fuel at its core. It will begin to grow again, slowly at first, but then quickly, as its dying core collapses.
The sun's surface will cool down, but unlike in the past, its larger area will outbalance its lower temperature. Earth will heat up, and the oceans will boil away. When the sun is forced to use heavier elements (carbon, oxygen, etc.) in its nuclear fusion reactions, it will bloat up to the size of Earth's orbit- swallowing and destroying our home, if we haven't found a way to move it by then. Don't worry, we have billions of years to work on this.

These are very good questions indeed. I will talk about the moon first; It is true that the moon is slowly orbiting farther from the earth -- and if you run the process backwards far enough you'd have the moon hitting the earth. There are several theories of where the moon cam from and one of the current leaders is that the moon was actually created when a large asteroid hit the earth billions of hears ago -- after the earth had formed, but during the process of planet building (i.e. when the solar system was still forming). Since heavier metals and stones would sink in the largely molten mantle of the earth, a large impact could scatter a large mass of the predominantly lighter mantle material into orbit. When Apollo visited the moon some twenty years ago the samples they brought back consisted largely of aluminum and titanium silicates-- later density studies confirmed that the moons complement of iron and heavier metals is lower than the earths by proportion. So -- effectively, it is possible the moon was in contact with the earth... The sun's size is set by several competing processes -- how much heat is produced versus the density of the solar plasma. Over long spans of time the sun is slowly growing warmer (again a theory-- but a carefully tested one). Although its size is indeed changing, such change is not linear with time and is a very slow process. The solar system appears to be no older than about 4.5billion years, very early in this process, the sun would have reached a size (about 1 million miles in diameter) which is only slight larger than the 864,000 miles (through the equator) we measure today. -- However, if you'd like to wait a few billion years, the sun will grow to be at least as big as the orbit of Venus. It might sound odd, but it is true that we know more about the sun than the core of the Earth. This is because the sun is a much simpler physical system -- essentially a very hot, dense gas ball, vibrations on the surface of the Sun allow measurements of the density most of the way to the core... (Look up Solar Seismology on the net).

The moon is currently receding from the earth about 1 inch per year. You are growing at a rate bigger than that. The important thing is that even over millions of years the distance the moon has moved is relatively small compared with its distance from us. In other words if the moon was moving towards us at a constant rate of 1 inch per year, it would take billions of years for the moon to be next to the Earth. As far as the SUN ,it is 93,000,000. miles away and the change in the solar radius is tiny compared to its distance from us.

I believe the moon is moving away from the Earth at a rate of about 4 centimeters per year, so even in a billion years the moon will only move 40,000 kilometers away, which is just a fraction of the current distance of the moon. I think it is still not understood very well how the moon was formed, but it somehow started orbiting the Earth a long time ago and has slowly moved away since then due to the same (gravitational) forces that cause our high and low tides on Earth. By the way, since the moon is moving away from us, eventually there won't be any more total eclipses.
Also, the days on Earth are getting slightly longer which is good news for those of us who always complain that there is never enough time in the day.
I hadn't heard of the Sun shrinking, but the theory of the formation of the sun is that a big gas cloud slowly collapsed over millions of years to become the Sun. The planets are made out of remnants of the cloud that were left behind during the collapse. So the Earth formed only after what would become the Sun shrunk enough so that the Earth wouldn't burn up. That's not really the end of the story though. Unfortunately, when the sun exhausts all the hydrogen that it uses for fuel, it is going to expand again as a red giant and the Earth will get burned up. We've still got about five billion years until that happens so I don't think anyone is too worried about it yet.