Learn English – How to parse “get done doing something”

In this context, way = direction, so it means we sent ten cans of yams in his direction, towards him.

It's an informal idiomatic expression which can always be replaced by to [whoever], but sometimes that underlying "literal" meaning imparts a slight nuance. For example, you might be slightly more inclined to use it if you're diverting something that would otherwise have been sent somewhere else. Or if the recipient/location/route you're sending it to/through is somehow "non-standard" for the thing being sent.

Off of in that context means from.

So the children are copying from one another and the teachers are trying hard to stop it.

Off of is quite informal. It is being used here because it's meant to be a child talking or narrating.