Assume the following list of functions and arrange them in ascending order of growth rate with proper justification through analytical reasoning.

g1(n) = 3n^4 + 6n

g2(n) = n * log(n^1000)

g3(n) = 7n^3 * log(n) + 1000

g4(n) = 3^n

g5(n) = 6^n

g6(n) = 1024n^2 + 4n + 460

Further, plot all these functions and identify if there are any cross-overs between them. If there are any cross-overs, determine those points at which the growth of the corresponding functions change. Please show an image of the plot and explain step by step.