I'm imagining a system with a star and something like 6-8 planets. The planets' orbits are (relatively) close to each other, and all share an orbital period that is exactly the same. I.e. the length of a year would be exactly the same on all planets. I've even gone so far as to imagine a system where the planets are all in a line, with a gravity elevator linking each planet to its neighbor(s). Something like this:
I understand that they would have to be offset a bit (or in slightly different orbital planes) in order to not perpetually eclipse each other. Assume that technology exists allowing a species to exactly place a planet into the desired orbit, and even make routine corrections (though I'd prefer to not have to if possible). In other words, you can almost treat each planet as a giant spaceship as long as it would not need to use any thrust 99% of the time.
In this scenario the planets would all be roughly Earth-sized and have somewhat Earth-like climates, though probably the innermost would be hotter and the outermost colder.
From what I understand, a planet orbiting at x (average) distance from the star has a specific range of velocities it must adhere to--if it is too slow it would crash into the star, and if it is too fast it would escape the system altogether. I also know that the innermost planets would travel slower and the outer ones much faster in order to make one revolution in the same period. Finally, I'm guessing the distance between each planet would vary throughout their orbits since the orbits would be elliptical (so the gravity elevators would be long and flexible). But I don't understand the math enough to do the calculations.
So specifically I'd like to know:
Is is possible for such a system to exist?
If so, are there are limitations/constraints on the length of a year in this system, or on the type of star, distance from the star, etc?
How far apart would the planets have to be in order for the gravity of each (assuming Earth-like mass) to not pull its' neighbors out of orbit?