Yes you can, the walls spring into existence, therefore anything within the cube that's is blocked by the walls is simply stuck inside until the cube is deactivated.

Ref. DMG pp159-160

... Otherwise a barrier of invisible force springs into existence, forming a cube 15 ft on a side. The barrier is centered on you, moves with you, and lasts for 1 minute...

No (mostly)

Breath attacks that aren't based on gas or wind would be unaffected. A red dragon's flaming breath attack is not harmful because it's windy or gassy, it's harmful because it's fire. Similarly a blue dragon's lightning “breath” and a black dragon's acid breath are harmful because they spit streams of lightning and acid, not because they're blowing on you.

Breath weapons based on gas would be affected just fine though: a gorgon's petrifying breath or a green dragon's chlorine gas breath are both harmful gasses, which the Cube of Force could block. You can tell because their descriptions call them “poison gas” and “petrifying gas”, respectively. Similarly, the Cube would keep you safe from an air elemental entirely because it is made of wind.

If you want to block non-gasses, use the other faces of the Cube — that's what they're there for. A gout of acid would be blocked by face 2, and something like a red dragon's fire would be blocked by face 5. Sure, they cost more charges, but the Cube is self-charging, so there's no need to lawyer-up (or worse, suddenly assert that D&D is science fiction and science-up; you do not want to dive down the science rabbit hole when dragons are involved, nevermind the lesser but still Gordian metaphysical knots you'd tie yourself into trying to define the distinctions made by the Cube's other face's effects in scientific terms) just in order to save a couple of charges you'll get back soon anyway.