1. Technically rivals aren't limited to 5, EX3 Leon has 15(!) iirc. But yes, 5 is the max you can have per character without regenerating them with Azusa's burst skill.
2. The only events that give song rewards are Allstar Lives and EX1-3. Rank ups give skill points and unlock more of the skill tree.
3. The amulet effects are straightforward, they increase fans, money, or EXP gains. The pandas increase fan gain, the cats increase money, and the birds increase EXP gain. There's three versions of each: small, medium, and large (they line up in that order if you have subsequent versions).
4. I've never bothered trying to get all of the gifts from events, so I don't know.