Can a moon be nearly as big as the planet it orbits?

Suppose the Earth’s moon is actually, say, a copy of the Earth itself. So there are two Earths revolving around the Sun.

– To what extent is life possible on both Earths? (Isn’t there any problems with gravity or something?)

– Can both Earths be the same size?

– Or should one always be smaller than the other?

– Have such systems already been discovered?

– Is this also possible with two gas planets?