Yeah insertion pipes (and RP tubes) don't work right with reactors because of its variable inventory those add-ons don't know when it's "full".
As far as filling up the chests with the overflow, I figure it doesn't really matter so you can just run stone pipes up to all the chests and eventually ice will get there (even though it'll choose the closer ones first) and it'll bypass any full chest and keep moving.
I have a system for filling up the chests with the overflow that I made.
All the overflow goes into several chains of distribution pipes that make sure that each of the 32 chests gets 1 ice in turn. That way no chest will have tons more ice than others, and it ensures that none will overflow, and they will have enough ice to go a full cycle because none of their ice is being wasted.