If you supply the hive with a berry, you come out of a random hole each time. It is not based on adjacency or any other rules other than random number generation as far as we are aware.
The exception, as of the Gemini Update, is the Split Juniper map. Unless there are no holes left, you will always come out on the same side of the map that you went in.
source @Viotech3#8512 on Discord
Yes, throwing counts for berries. You know how it has a flashy trail behind it? As long as it has that, it counts, even if it rebounds. The only time it doesn't is when they get kicked or boinked around. Even if you're the one responsible, it won't give points.
source @Sykes#7290 on Discord
Nope, see next section.
When all the berries are destroyed/consumed, it's called a famine. Berries do not grow back, so there is no undoing a famine. There's also "soft famine", which is when there are few enough berries remaining that economic victory by either team is impossible, but the map is not completely out.
This effectively cuts off the economic route to victory, and you are now left with either military or snail victory routes only.
Helix Temple, notably, has less berries than any other map, and so is more likely to have famines (although Spire is notorious for having all the berries pool at the bottom where they're hard to access in long matches, it rarely has actual famines).
Just kidding, they're berries per the developers.