TL;DR: Nothing like you describe is possible with current technology (or anything likely in the near future), but several configurations are possible using technology well within the realm of science fiction.
As others have mentioned, the linear arrangement you mentioned is not possible (unless the connections between the planets are exremely strong and rigid). As others have also mentioned, a Klemperer rosette would allow the planets to maintain fixed relative distances, but the orbits would not be stable to small perturbations. The Klemperer rosette also has the possible disadvantage that the distance between the planets is similar to the distance between the planets and the central star (i.e. quite large).
There is a another configuration worth considering, although it doesn't solve all of these problems: the proposed orbital configuration for the Laser Interferometer Space Antenna (LISA):
In the proposed LISA mission, three satellites orbit the sun with roughly the same orbital period as the Earth. Their orbital planes are oriented at slightly different angles and their orbital phases are synchronized so that the three sattelites orbit the sun while maintaining a fixed relative distance. As seen in the picture, the constellation of sattelites also rotates in the plane defined by the three satellites. Note that the satellites are not attached to each other in any way (the lines on the diagram indicate laser beams used to measure the stretching of space).
While this keeps the planets relatively close together, I believe it suffers from the same instability problem as the <PERSON> rosette, so while it works for satellites which are small and far away from each other, the gravitational effects between the planets in a LISA-like configuration would be considerable, and would lead to significant instability. The interplanetary gravitational forces would also cause the planets to be non-periodic, although not in a way that would neccesarily present any practical problems.*
So, to answer your questions:
1. It is not possible for a system like the one you describe to exist without some repeated adjustments of the orbits, or extremely strong connections between the planets. However, some configurations require more energy to maintain (or stronger connections) than others. The <PERSON> rosette and the LISA configuration both require orbital adjustments only to fix any drift away from their initial configuration, so in principle this could be done with a relatively small amount of energy or relatively weak interplanetary connections.***
2. 801
According to <PERSON>'s third law, the length of year, distance of the planet from the star, and mass of the star cannot be varied independently, although there is no restriction on any one of these parameters independently. In principle**** you can pick whatever you want for any two of these parameters, but your choice of the first two determines the third parameter. This is still true for the <PERSON> rosette and the LISA configuration, although the exact relationship between the parameters would be different.
In the <PERSON> rosette, the gravitational forces between the planets would shorten the length of year for any given mass of star and radius of orbit, although the effect would be very small unless the planets were very massive, or the star very small.
In the LISA configuration, the time it takes for the planets to orbit the star would be about the same as you would expect from Kepler's third law. Depending on the rotational axes of the planets themselves, the seasons could change in very complicated ways since the season depends on the orientation of the planet's rotational axis with respect to the incoming starlight. This in turn depends on the orientation of the rotational axis relative to the orbital plane of the planetary constellation, as well as how the constellation is rotated in its own plane. Since the rotation of the constellation has a different period than the orbit of the constellation around the sun**, the angle between a planet's rotational axis and the incoming starlight could change in quite complicated ways over cycles of many (orbital) years.
3. The gravitational force between two objects is proportional to $1/r^2$, where $r$ is the distance between the objects. This means that planets will feel each other's gravitational forces no matter how far away they are from each other. In the case of the <PERSON> rosette and the LISA configuration, these forces will affect the orbits (see the previous two paragraphs), but these effects are not catastrophic. The bigger problem is that these configurations are unstable, so the gravitational effects of everything else in the universe will be catastrophic in the long run. It is these effects that you have to counteract using orbital adjustments or strong interplanetary connections. Luckly, most of the rest of the universe is very far away, so these effects are small and, with the right sci-fi technology not terribly hard to fix. 24