My $0.02 worth...
To your last question, imaginary time, yes this is a mathematical convenience.
The use of the word "straight" here is not quite right. The analogy is this picture you and another person walking south from the north pole starting in slightly different directions. You both begin moving away from each other but by the time you reach the equator you're distance apart stops growing and starts shrinking until as you reach the south pole you meet. You are walking "straight" in the sense that you neither turn left nor right but your paths are curved in the sense that the distance between you doesn't change uniformly. This is because the surface of the earth is not flat but curved... i.e. curved 2-dimensional space.
In SR space and time are components of 4-dimensional space-time so you can imagine objects traveling at different velocities as traveling in different space-time directions (at a rate of one space-time unit per proper-time unit.) In GR this space-time is curved so as you "walk" in space-time your direction=velocity can change. You are accelerated relative to other object's paths.