You actually need to equip the sword in Shift in order to upgrade it. Not sure why you can unequip it, though my hypothesis is that there was originally plans for more than one type of weapon (like in Alharan where there's swords, spears and daggers). Also, the white laser beam is Storm. The other two spells are Wind and Flare. While Storm is under most circumstances the best spell in the game, Flare has less of a blind spot, has similar DPS and is easier to hit with, in exchange for being basically melee range.
While it's not like the plot is entirely hintless (at least in the original Japanese), hints are so scattered and sparse you'll probably have almost as much luck progressing the plot just putzing around randomly. On that, I agree that the developer probably could've added more/better hints to what the townsfolk say or something. I also can't deny that most of the plotlines don't provide much in the way of rewards (aside from the oni girl's plotline, because that unlocks the sword's final upgrade level). On that, I agree that it wouldn't hurt to add some more loot at the end of those plotlines.
By the way, skill EXP is earned from defeating enemies with the sword/magic, taking hits, or performing thievery-related acts (e.g. lockpicking). If you're training magic you can soften up an enemy or three with the sword and blast 'em all away with one magic spell. So long as the last hit is from your magic you get full magic skill EXP.
