Antonym of the given word

abet 

An odometer is an instrument which is used to measure Mileage of a vehicle. In the same way, Compass is an instrument used to know the direction.

Hence, Odometer : Mileage :: Compass :: Direction.

A Bill becomes a Law; a Larva becomes an insect. (Or)

A Bill turns into a law; a Larva turns into an Insect.

To rotate is to energy to stop is to inertia. From the definition itself we know that inertia is a property by virtue of which a body continues to be in the state of rest or in state of motion. So over here whenever energy is supplied then the body starts to move that is a displacement is there and it is rotating where as whenever the body is stopped an inertia of rest is created.

So, Rotate : Energy :: Stop : Inertia.

Lethargic means "sluggish", kind of inactive

Vital means full of energy, lively

So, they are antonyms.

So, for the 2nd part of the analogy, you are looking for a antonym to trite

Trite means lacking originality or freshness. Overused. Kind of like a cliche

So, an antonym could be original, unique, innovative.

A Library consists of Books in the same way Book consists Pages.

Hence, Library : Book :: Book : Pages.

Warm and hot are in the same meaning sense , so not so good is same as worse in meaning.

A sponge is a porous material. Rubber is an elastic material. 

Since Rubber is not a Massive or solid and not an inflexible thing, it is elastic. 

Here Grass is to Erosion and whereas Dam is to Water.

Since Grass prevents Erosion similarly Dam prevents Water flow.

Hence, Grass : Erosion :: Dam : Water.