Considering Fa Jin's meaning, it should've only lasted for that single jump, the energy around his leg vanished as well. Though he may have used Fa Jin again, as his legs glow again before the flashback. However it's best to wait and see before speculating.
Also of course he's using Danger Sense, it's not like he can turn it off. Just because we don't see the Danger Sense signal, doesn't mean it's not activating. Especially since he dodged a bullet from behind once again, he wouldn't have notice it if Danger Sense didn't tell him.
The bullets are indeed faster than him, however it's not by an extreme amount.
Izuku dodged that shot to his stomach before she fired it. The distance he moved is far less compared to the bullets, he would've been hit had he moved at the same time. Nagant can't fire the bullets at the same time, he's dodging those shots one after the other. It's very impressive, but he can be slower than the bullets and still dodge them.
Also I've been planning on making
Nagant's profile, though it isn't ready yet, only a rough draft right now. At the least I'm waiting for the battle to finish.