Is there any relation between atomic vibration and atoms in excited or ground state?

I'm going to assume you mean molecular vibrations because single atoms don't vibrate unless we're talking about lattice modes or something. Anyway, there is no guarantee that the vibrations in the ground state of a molecule will be the same as the ones in any random excited state. This depends on how great the change in geometry is between the two electronic states. You can easily experimentally verify that there are excited states which have the same modes as the ground state using Raman spectroscopy. Even though that it is not usually how people think of Raman scattering, it is the result of a three state interaction involving an excited state coupled to the ground state's modes. Just any excited state though, you have to do more work to find out which one and a mode not being Raman active also doesn't ensure there isn't an excited state without that motion.

How accurate is the statement that all energy on Earth originates from the Sun?

I think that this statement is inaccurate considering that the sun with all planets were formed from a nebula cloud of gases and particles which were formed from spacetime fabric. The sun is just part of that nebulae! As far as food is concerned, the sun is the source energy for its organization.

What will happen if we do not look after the Earth?

The Earth surface will become ocean, welcome to water world. Unsustainable, especially to human life, if not it's guaranteed you'll live in a houseboat and maintain a seaweed farm.I'm trying to prevent that from happening, but it will mean many changes in human behavior, and they are mandatory, crucial, not