The Gold Skultullas unlock extra battles when you collect a full set, and you unlock new costumes in those stages. They don't give combat related power ups.
You also get hearts by leveling up characters, so, you shouldn't need to worry about them regarding your characters' powers, they're mostly just for 100%. In fact, when you start playing through Adventure mode, where you need to achieve specific rankings in the mission, the extra hearts are mostly useless, since in most missions you won't be able to get A rankings after losing more than 8 Hearts, doesn't matter how many hearts the character actually has in total.
The character has to be an npc (or a playable character that you temporarily can't control due to a mission specific event) and their health needs to fall after a certain point, which is likely around 30 or 40%. There will be a green circle around them, indicating that they can be healed when you approach them.
Each weapon type has 4 different weapon variations, each one with increasingly larger damage output and skill slots.
For example, for the Hylian Sword, Link's first weapon type:
Knight's Sword -> White Sword -> Magical Sword ->Magical Sword +
Within each type, there are also stars. The more stars a weapon has, the more damage it causes. The stars go from 1 to 5. Note that a 5 star Knight Sword will still be weaker than a 1 star White Sword, and the same is true for all weapon types. When you get a new weapon from a chest, or from getting a new character, it'll come without stars and skills, but then the weapon will be able to appear as a random drop from enemies, which is where you'll get stronger versions with more stars and skills.