What determines how far a telescope can see in the universe? How does recording data for a very long time (~10 years) help? If we could build a telescope which work at microwave region, will it be a better telescope (so that we could see even farther)?
We do have space telescopes (COBE, WMAP, Planck) that operate in the microwave region, and indeed they do see much, much deeper into the universe than HST. The Hubble Extreme Deep Field renders images of a universe that is a few hundred millions of years old. In contrast, COBE, WMAP and Planck provide us with a view on a 0.37 million years old universe.
This doesn't mean that COBE, WMAP or Planck are 'better' telescopes than HST. It's a matter of different tools for different tasks: the microwave instruments observe the cosmic microwave background and in doing so tell us a lot about cosmology and the Big Bang. The Hubble observes galaxies, nebula and star systems that formed much later.