Does having something frozen near you displace the hot air around you?

Yes the cold would force hot air to rise higher and away from the ice.

Larger surface area of the ice would displace more, due to a larger surface cooling the air.

Ie: 50mm square ice would displace a small amount of hot air.
Were as 50cm of square ice would displace hot air faster due to area.

But it would displace at the same rate per mm square regardless of size.