The Green Dragon's breath is described as:

Poison Breath (Recharge 5-6). The dragon exhales poisonous gas in a 90-foot cone. Each creature in that area must make a DC 22 Constitution saving throw, taking 77 (22d6) poison damage on a failed save, or half as much damage on a successful one.

It is "poisonous gas" the Dwarf gets advantage on the save.

Poison. Venomous stings and the toxic gas of a green dragon’s breath deal poison damage.

It is "poison damage" the Dwarf takes half damage (effectively 1/4 if saved 1/2 if not).

With respect to your comments on protection from poison, I think you are splitting hairs: it is poison so advantage is given on the save; the fact that the effect is to do poison damage rather than impose the poisoned condition is irrelevant to the save.

That said, the DM is the sole authority on rulings so, check with your DM.

No.

The key factor is:

It hedges out vapor, gas, and fog that can be dispersed by strong wind.

So if the effect description of said gas includes a note that it can be dispersed by a strong wind, it will be.

The green dragon breath has no such statement, therefore it is not warded by Warding wind.

OP asked in comments if a stronger wind would affect the Green Dragon's Breath:

It doesn't matter the wind. The breath of the green dragon would spread out to its cone-shaped area of effect even inside a hurricane. Unless an effect's description states that no gas can get through or something amounting to that, it doesn't matter. That spell cannot stop the breath because the breath cannot be dispersed by wind (and the spell states that it only affects what "can be dispersed"). The dragon entry would have "can be dispersed" if it would.

Regarding wind as weather in the DMG, a strong wind would not affect the Dragon's breath, because it is gas, not fog:

STRONG WIND

A strong wind imposes disadvantage on ranged weapon attack rolls and Wisdom (Perception) checks that rely on hearing. A strong wind also extinguishes open flames, disperses fog, and makes flying by nonmagical means nearly impossible. A flying creature in a strong wind must land at the end of its turn or fall.

User Doval in the comments below brought an example of a higher (3rd) level spell without the that can be dispersed limiter:

(...) Wind Wall also creates a strong wind and says: "The strong wind keeps fog, smoke, and other gases at bay" which RAW would keep out a green dragon's breath. (...)

An effect that stated "blocks gases" without the "that can be dispersed" remark of Warding Wind would be effective to block the Green Dragon's Breath.

Example of an effect that blocks gases without the "that can be dipersed" clause:

Cube of Force

Face 1, 1 charge: Gases, wind, and fog can’t pass through the barrier.

Example of an effect that is hedged out by warding wind:

Fog Cloud

You create a 20-foot-radius sphere of fog centered on a point within range. The sphere spreads around corners, and its area is heavily obscured. It lasts for the duration or until a wind of moderate or greater speed (at least 10 miles per hour) DISPERSES it.