Will my case LED lights normally light up without motherboard/HDD, just PSU?

no. you need a motherboard. even tho the power supple is turned on it will only supply a standby power to the motherboard connector, not to the 12v and 5v connectors. there is a pin on the motherboard power connector on the power supply which the motherboard uses to tell the psu to turn on fully when you press the power on.