Everything must be on a grid for the Challenges, so yes they would "hop" from tile position to tile position like they are on a grid. if you dont want to do the Challenges (you wont get as high of a score at the end) you can be as free as you would like with the tiles so you could move them wherever you like, its really just up to you. :)
And yes the color per tile rules are much like how the ZX Spectrum (or other systems from the time period) worked.