For Problem FF-1:

As the problem suggests, when you put the red dot on an eigenspace you get a periodic solution, with the period depending on the eigenvalue. One eigenspace has a shorter period than the other.

The idea is that any initial condition (initial position of the red dot) can be expressed as a linear combination of two linearly independent eigenvectors. Then as time goes on the solution is the same linear combination of the two ``eigensolutions''. Since they have incompatible periods the motion never repeats. (Periods are compatible when their ratio is rational, in which case the motion does repeat.)

For Problem FF-2:

(a) Usually some initial conditions will give more lively motion than others. The solutions will not generally be periodic unless you set the initial conditions to an eigenvector. Nevertheless, each solution is a linear combination of periodic solutions, and the largest coefficient might be for a fast periodic solution or a slow one, giving different looks to the combined motion.

(b) (Nothing to answer.)

(c) Yes, it will normally affect the period.