does it have a logo on it?

Yes a small grey/white logo that says dry sweat. But you can turn it inside out and you won't see the logo or know that's it's turned the opposite way

Will this keep my edges tame after an intense/sweaty workout? Im looking for something to pull sweat off my hairline and keep my edges laid post WO.

yes it will keep you edges down as far as sweating off your face it will help but not completely sweat free. If you are a heavy sweater it will help.

Can you wear this under a hat? When I'm outside, I still need sun protection too!

Yes you can wear this under a hat. It is fully functional,it is not thick at all.

I have eyelash extensions and am wondering if these headbands help protect the sweat from dripping onto my eyelashes?

It didn't help me cuz when I would sweat the band would slip off my head. I'd recommend u find a headband from an athletic apparel retailer with secure velcro