Ask An Astronomer 
Here are the answers to selected questions. Questions may have been edited for clarity or brevity. Click on a link to move directly to the answer.
That's an interesting idea, and while the scale of our Galaxy is not quite the relevant one, this same question does arise on the much larger scale of the universe as a whole.
As we have learned how to build larger and better telescopes, astronomers have been able to detect fainter and fainter sources. Some of these sources are just intrinsically less luminous (for example Kuiper Belt objects at the edge of our solar system), but mostly these sources are so faint because they are really far away.
Light travels at a finite speed, and so when we image very distant objects, we are actually seeing the objects as they looked like many thousands (or millions, or even billions) of years ago. It just took the light that long to travel from the source to our telescope mirrors here on Earth.
Alright, that much you already knew. Now, our galaxy, the Milky Way, is almost certainly rotating. We know this from detailed studies of the molecular hydrogen gas that we detect all throughout the disk of the Milky Way. However, some geometry will quickly reveal that it is impossible for us to see an image of ourselves emitted when our solar system was on the opposite side of our Galaxy many years ago.
Let's approximate the orbit of our Sun around the Milky Way's center by a circle. The radius R of this circle is the distance from us to the center of the Milky Way, equal to about 24,000 light years (or 140,000,000,000,000,000 miles). The distance between us now (call this point A) and the point on the opposite side of the circle (point B) is 2*R = 48,000 light years. This means it takes 48,000 years for light to travel from A to B. The Sun itself, however, is traveling on a circular orbit, it has to go half the circumference of this circle to get from A to B. The circumference of a circle is given by 2*Pi*R, so half the circumference is about 75,000 light years. In order for us to be able to see an image of ourselves emitted when the Sun was on the opposite end of its orbit, the Sun would have to travel these 75,000 light years in less time than it took the light to cover the distance. In other words the Sun would have to be orbiting the center of the Milky Way at about three times the speed of light to achieve this - an impossibility, as you know.
On the much larger scale of the universe as a whole, the possibility of observing a past image of our own Galaxy has been considered. If the size of the universe is finite, then light would take a finite amount of time to travel the entire length of the universe. If the age of the universe is older than this time, then in principle it should be possible to detect an image of ourselves, emitted at a much earlier time. How could the size of the universe be finite, you might ask. Well, if it has non-zero global curvature, for example. What if we lived, so to speak, on the surface of a big old bowling ball? The area of the Earth is finite, yet it appears boundless to our unaided human eyes. Another, more playful, example is the computer game Pac-Man. If you leave through one of the portals on the right side of the screen you pop out on the left side. This might sound silly, but some cosmologists have been considering topologies similar to this in recent years. If you can image three-dimensional space itself as being curved, then you can perhaps allow for the possibility of light leaving our galaxy several billions of years ago, streaming away from us in all directions for eons, and eventually completing one passage through the entire universe only to be observed by our own eyes today.
Observational evidence tends to disfavor such notions, but it's not completely ridiculous.
This, it turns out, is a hotly debated topic of current research. Over the past ten years or so, a somewhat coherent picture of galaxy structure has emerged. Slow, steady cannibalism of small galaxies is thought to result in spiral galaxies like this one.
Violent mergers between equal mass galaxies is thought to produce elliptical galaxies like this one.
However, numerical simulations of the formation of galaxies in the early universe indicate that slow, steady cannibalism of small galaxies tends to smear out coherent rotation (and possibly make galaxies look like formless blobs, i.e., ellipticals?), while violent mergers tend to produce coherent rotation (and possibly make rotating disks of material, i.e., spiral galaxies).
Unfortunatly, it's not clear what's going on. The most detailed computer simulations aren't detailed enough to be confident that they produce the right answer, and galaxies evolve too slowly for us to watch what happens in the real universe. So there's no clear answer to your question!
The current wisdom is that these planets formed much further from the star (at distances comparable to where Jupiter is found in our solar system) and then migrated to an orbit very close to the parent star. This can happen, and is expected to happen, if there is still a significant disk made up of gas and dust when the planet forms. Obviously, there must be some gas and dust or else there wouldn't be any raw material to make the planet itself; it's a question of how much gas and dust there is when the planet forms.
The best evidence for this line of reasoning comes from one star called HD 209458, which hosts a 0.6 jupiter mass planet that happens to pass in front of the star from our perspective. Thus the star dims slightly when the planet blocks some of its light, and the amount of dimming allows a very precise measurement of the radius of the planet. It turns out that the radius is that of a gas giant planet, not a rocky one.
However, this information is only for one of the hundred or so extrasolar planets currently known and it's dangerous to draw conclusions about many objects from information about one object. Thus, I should mention that people have thought about your suggestion: Is it possible to form gas giant planet so close to the star? Is it possible to form rocky planets that big? Rocky planets that close to the star? These are all possibilties, it's just that at the moment the scenario I outlined above seems the most plausible.
Thanks to Jenny Graves, Nick Konidaris, Chanda Prescod-Weinstein, Laura Chomiuk, Ryan Montgomery, Greg Novak, Mike Kuhlen, Kathy Cooksey, Scott Seagroves, and Sally Robinson.