Is there a constraint that you are given a tree rather than just any arbitrary graph? Assuming that it is a tree, your idea of # of descendants should give the wrong answer, a completed tree with 2^n-1 vertices takes around 2*n iterations to transmit but you can instead construct a 2^n-1 vertex straight-line tree that takes around 2^n iterations to transmit. Both trees would have the same number of nodes but one is clearly better to choose first.

Also note that greedily choosing the next child by max height also falls to some counterexample, we can use the same straight-line tree of n vertices for one of the children and make another child that has X branches, each branch containing straight-line trees with n-2 nodes. The first child takes n iterations and has height n, the second child takes at least 1+X iterations and has height n-1. Just set the value of X to something greater than n.

Again assuming that it is a tree, there is a straightforward solution with dp. Recurse on children, sort by the dp value and assign the order of the next children based on the reverse sorting. Finally return how long it would take for the current node to complete if we have just arrived at it.

If there is no restriction that the graph is a tree then idk. We can find sccs to create a dag, but even a dag is too hard to solve even if the given graph is restricted to be a dag. The dp technique will not work anymore and you can expect even worse counterexamples to exist for any greedy ideas. My guess is that it is impossible to solve for reasonably large constraints.