Can the Hubble telescope bring any star into focus? Lets say I am talking about a view like this supernova - 13 billions light year away. In short can Hubble bring any star into focus in the entire universe? And if so, to what definition? 
I also wonder, how much time would time would it need to focus on a distant star or planet, in comparison to a closer one?
 A: Telescopes and any other optics that look at objects in space are set so that the focus is at infinity as defined by the equations of optics.
Resolving power has to do with the size of the primary mirror and the wavelength of light that you are interested in.  The smaller the mirror, the less angular size you can resolve (so you can only separate two objects that are far apart).  The bigger the mirror, the closer two objects can be and you can see them as separate features.  Similarly with wavelength, the longer (redder) the wavelength of light that you're interested in, the bigger the mirror you need, whereas the shorter (bluer) the wavelength, the smaller the mirror can be.  This is known as the diffraction limit.
But there's another factor to consider, and that's the resolution of the actual detector.  Let's say that your diffraction limit for 500 nm light (green) on a 2.4 m mirror (Hubble) is roughly 0.05 arcseconds.  But, if your detector only records 1.00 arcseconds per pixel, then that is your limit.  Usually detectors are designed to be at or a little better than the diffraction limit of the optics, though.
In terms of time needed, that has to do with how bright the object is per angular unit.  For example, the Andromeda galaxy is very bright as a whole, but that brightness is spread over a large portion of the sky.  So the brighter the object is and the more concentrated that light source is, the shorter time is needed to properly photograph it.  And this also has to do with the "speed" of your optics -- a larger f/number is "slower" and requires more exposure time than a smaller f/number.
But every detector is different in terms of how sensitive it is and how long you actually need to record the same amount of light.  This gets into issues of quantum efficiency that I think are beyond your question.  Suffice to say, there is no set equation to know how long you need to expose an object to properly capture it.
A: I think what you are asking is if the Hubble telescope can "resolve" any star within 13 billion light years.  The answer is "no".  
Resolution is the measure of telescope performance that determines how small a detail can be resolved in the image it forms.  Nearly every visible star resolves to a point of light in the Hubble or virtually any other telescope.  Betelgeuse is an exception that has been resolved as a disk.  It is both very large and relatively close, at only ~640 ly.
The stars in distant galaxies are hard to resolve because they are too close together to be separated.  Same with stars in the Milky Way that are close together but where one is considerably brighter than the other.
A: Once a telescope is set in proper focus, objects at all distances are rendered into sharp images without any further changes in focus settings. However, the lowest luminosity of stars that can be observed does depend on distance. And the farther away a galaxy is, the less detail can be seen in its image.
Hubble can image a star like the Sun about as far as the nearest spiral galaxy, M31, which is about 2.4 million light years away. If the Universe were unevolving and unexpanding, a supernova could be observed by Hubble up to 30 billion light years away. But the Universe expands, shifting visible and near ultraviolet light of the most distant stars to wavelengths longer than those that can be detected by Hubble instruments. Light emitted in the far ultraviolet would be shifted to visible wavelengths, except that this light was absorbed in the very early universe by intergalactic clouds of neutral hydrogen. 
The upshot is that the most distant objects visible to Hubble are about 13.2 billion light years away. And no telescope can ever see more than 13.8 billion light years away because that is the distance travelled by light since the Big Bang which occurred 13.8 billion years ago. As for the detail on these most distant objects, it becomes difficult to impossible on Hubble images to separate the image of a supernova from its host galaxy.
A: Telescopes do not focus. The objects they view (even the Moon) are so far away compared to the aperture and focal length of a telescope that they are all at infinite focus, so telescopes use a fixed focus.
Whether a telescope can resolve a distant object or not is dependent on aperture and brightness. Brightness in turn is dependent on intrinsic brightness and distance. Hubble can resolve even incredibly distant objects if they are bright enough, but cannot see any object at any distance.
