Everything is energy...matter is energy.
Technically yes, but no. It can be transformed to energy without loss, and it exerts itself on the SEM tensor, but strictly speaking matter is not energy. Mass is, though, if that's what you mean.
Everything is electrically charged, including us.
Again, yes but no. There are several particles that aren't electromagnetically charged, photons and neutrinos for instance. Any force carrier, actually, and possibly a few more that slip my mind. It could easily be noted that photons, while not electromagnetically charged, are still very slightly affected by gravitation. While we haven't observed any other particles having a similar effect, this can be attributed to our lack of ability to detect neutrinos at long range.
So by saying, "gravity is the force that attracts a body toward the center of the earth, or toward any other physical body having mass", it doesn't explain WHAT the force is. It's too general.
See General Relativity, the current gravitational theory. It explains what gravitation is.