From the title I'd guess you're thinking about the reports that gravitational waves had been detected by the BICEP2 team. They measured the polarisation of the cosmic microwave background and found patterns that they concluded must have been caused by primordial gravitational waves during inflation. However it now seems likely that they misjudged the background noise in their data, and it's no longer clear whether they did detect GW or not. If anyone's interested their paper is here, or Google BICEP2 for many, many blog articles on the subject.
So at the moment the only secure GW detection is the indirect detection from measurements of the Hulse-Taylor binary, as Dave mentioned in his answer. However this doesn't invalidate your question and we can still address the points you raised.
Why GW requires such energy levels?
Cosmic inflation happened early in the evolution of the universe when energies were enormously high, and (assuming our theories are correct) it would have generated copious amounts of gravitational radiation. But gravitational waves don't require a high energy. As an analogy consider creating sound waves: if you clap your hands it makes a sound people can hear from a few yards away, while if you let off a hydrogen bomb it makes a sound people can hear from miles away! It doesn't take high energies to make a sound, but high energies make a very loud sound.
In principle you and I generate gravity waves just by moving around, but the intensity of these waves is absurdly small. The Hulse-Taylor binary is vastly more energetic than you and I, but even that generates such small amounts of gravitational waves that it requires exceedingly careful measurements to detect them. It isn't that the very high energies involved at the time of inflation were needed to generate gravitational waves, but rather that such high energies were needed to generate gravitational waves that we can still detect 13.7 billion years later.
Why attractive gravity doesn't create GW?
It isn't the attractiveness or otherwise of gravity that matters. Gravity waves are created by any distribution of matter or energy that has an oscillating quadropole moment. This would apply to exotic matter and negative matter (if they existed, which they probably don't) in the same way it applies to normal matter.
If GW supports Inflation Theory, why the exponential growth of the universe stopped (or did it ever stop?)
No-one knows the answer to this because we don't have a fundamental theory of inflation. The various theories of inflation postulate a scalar field that causes it, but without giving any concrete description of the physics that creates the field. Generally the field will be selected so inflation ends after the required number of e-foldings.
There are attempts to provide fundamental descriptions of the scalar field, e.g. linking it to the Higgs particle, but as yet no widely accepted theory has been found.