I believe she recieved some like naruto/kurama did with kakashi.
But that's just my theory.:y
When naruto went into BM mode, did he give hinata part of d naine tails chakra or she was just enveloped by naruto's BM mode
Why does she have one eye closed? Is it because Naruto is in a bright light?

I'm not sure if naruto can give chakra like that
Last time naruto switched to kurama when kurama healed kakashi
Yep, last time they had to switch, and although Kurama said that hes going to teach Naruto how to do that, its obvious that it didnt happen yet.