Before the seal obviously since they were portrayed to be fighting Kaguya.
:lol they both sealing up their mother with the seal on their hand, not sure how Hagoromo got the both seal after that to give to Naruto and Sasuke but it was implied that both sealed up their mother. The juubi and Kaguya were actually merged together as the manga implied :lol
There is no evident that Hagoromo has rinnegan before he got juubi chakra. Even then if he had it first, there are still possiblity that he has byakugan and then awakened rinnegan by any mean :lol
There is nothin to exclude that possiblity and also the byakugan has been inherited by every one else. There are also possiblity to support your idea that mutant was the key that Hagoromo got rinnegan instead of byakugan, but that is not very likely to happen. Because the very fact that presented by BlackZetsu that Kaguya found out that chakra being shared between her sons and she wanted it back. So if she saw rinnegan Hagoromo by birth then more likely she will choke Hagoromo to death while absorb his chakra, not waiting until he is a full grown up and try to steal it back by then :lol