Relationships

So I’ve got some questions.

First the daily gift, I don’t really get the daily gift giving when I try and give things to the villagers they decline it so what am I supposed to do ?

Second, should I be always fulfilling the weekly gifts for every villager or just the ones I’d like to pursue romantically?

Third, this isn’t about relationships but I’m so tired of running back and forth. Is there any way to be able to move faster ? Get places without getting tired of the game just being me running around from one end to another xD I love this game but I get so tired of running back and forth.

Fourth, again not about relationships but I haven’t been able to figure out the “ancient battery” quest. How do I solve the riddle ?

Fifth, not relationships but should I be focusing on story quests then deal with the villager quests ?