What's the meaning of "off" here? Is "eliminating ~ the planet" a subject of this sentence?
Good question! The subject of this sentence is not directly stated, and is actually indirectly referring to humans, as we would be the cause of an entire mosquito species being eliminated from the planet in this context. The preposition "off" in this context has the meaning of getting rid of something. To "eliminate off the planet" means to remove the mosquitos from the planet, or else, kill all of them until they are extinct. Here's some more examples of "off" in similar contexts: Ex: The flood wiped all the plants off the land. Ex: He is moving off the island.