Yep. I got the bronze bug badge fairly quickly thanks to the island, despite starting in winter. The fishing one will be a pain to get though as there's no reason to do much fishing.
Yes and it goes quickly if you replant the middle trees and add flowers. My side player goes in the afternoon and gets 100 bugs in a short period of time. Sell most to the little island girl.
Last night she got the silver amount caught badge.