I mean, in all of those circumstances I just hold my breath. I'm certainly not wanting to breath in a bad smell through my mouth.
But if the question is can I elect to breath through my mouth without pinching my nose, the answer is yes. I wouldn't describe it as "shutting my nose," though, so maybe we're not talking about the same thing.

The Legend of Korra spoilers:
Korra doesn't stakelessly get her bending back 15 seconds after losing it to Amon. They had the opportunity to give her the self-discovery arc that they tried to shoehorn into the post-Zaheer poison recovery.
The series could've had much better footing, in my opinion, if Korra had wrestle with that just a bit longer. Instead, a bunch of bald people show up and say, "I got you, Holmes," and then she's good as new.
It's been a while, and I've lost some of my steam over this. But yeah, that's the only one I've got the energy to type out right now.