Currently (Minecraft 1.17, Carpet mod 1.4.40), there is no way to query(...) or modify(...) the value of TicksFrozen in any way. A workaround is to use /data merge entity commands to modify the TicksFrozen value, but that does not work on players.

For example, I would like to make players and mobs slowly freeze to death if they are currently holding blue ice. The mobs part would work fine as I can just use commands, but the player part would require its own separate action under the query and modify functions to work properly.

noun (like 'fire') would be 'freeze', 'frost', 'freezing'?
immune_to_freeze? susceptible_to_freeze? -frost, -freezing

is_freezing is already good for is_burning antonym

For example,
query(e, 'is_freezing') checks whether the entity is currently taking freeze damage.
query(e, 'frost') returns the entity's TicksFrozen value.
modify(e, 'frost', value) writes value to the entity's TicksFrozen tag.