Would happen if the rate that Dark Energy is expanding the universe would increase, but if gravity increased with it so the great Rip wouldn't happen?

Remember Dark Energy is repulsive, the more the faster the universe expands (indeed accelerates), so the faster it would cool down and increase its size, not the other way. Now, no one knows what it is. For instance, if Dark Energy would rather be some other geometric effect, it could have a different ending than just if it is a cosmological constant term. So far, though, all experimental data do not favor anything else that is not compatible with just a cosmological constant ... We'll need to wait few years (~10 years) to get to know much better this with missions and ground experiments that are undergoing or planned to be launched.