Legendary Drops and Mayhem Mode Question

I just recently picked up BL3, but I’ve played all the BL games. I’m at the point now of farming legendaries, and I’ve seen youtube videos of people killing bosses and tons of legendaries dropping. I’ve never had that happen. The most I’ve ever seen drop is 4. I’m farming at Mayhem 3, do I need to bump it up to 4 to get the legendaries to rain like i’ve seen?

Also I’ve been told that upping the mayhem level does nothing to the stats of the weapons that drop. Is that true? If I get a legendary on TVHM it’s the same as if I get that legendary on TVHM Mayhem 3 or 4?

Thank You.