Are you able to use the HDMI and USB ports, while the hub isn’t connected to power?

The HDMI can be used even without power connected. As for USB ports, it depends on how much power your USB device consumes. If your device doesn’t use much power, like a USB flash drive, then you don’t have to connect to power; but if your device consumes a lot of power, like an external hard drive, then you’ll have to connect to power.