The Hubble constant is still a constant, but like most "constants" in physics, it is liable to change when we get better measurements or better information. To answer your questions,
1. Edwin Hubble came up with it in 1929 as follows. First he developed a way of estimating the distance to a certain type of variable star called a Cephid variable, whose rate of brightness variation is directly related to its absolute brightness, by looking at some Cephids near us (In particular in the star cluster in the Hyades where you know all the stars have the same distance from us because they are all in the same cluster.) He then found such stars in nearby galaxies. By measuring their apparent brightness, and knowing their absolute brightness due to the rate of variation, he could figure out how far away that galaxy was. He found that there was a rough linear relation (lots of statistical scatter) between the distance and the redshift of the light from that galaxy. That slope of the distance vs redshift was Hubble's constant. It is now expressed in terms of the velocity of recession in km/s, divided by the distance in MegaParsec (MPc).
2. It has changed every time a new measurement was made. Measurements never give the same value as previous measurements. In the case of Hubble's constant, this change has sometimes been large (Hubble, for example, was found to have gotten the wrong distance to the Hyades cluster which made his whole calculation of the absolute brightness of the Cephids suspect-- together with other problems).
In the 70s and 80s there was a long running controversy between Sandage and de Vaucouleurs who claimed the value to be 50 km/s/MPc and 100 respectively (with about 10% accuracy in each case). The latter result is problematic as the age of the universe (which is proportional to 1/H) is very small in the latter case.
The problem is how to measure the distance to distant galaxies. Can one accurately measure the distance to a large enough known sample of Cephids in our own galaxy? Can one accurately measure the brightness of the Cephids in distant galaxies. Can one find other measures of brightness of galaxies which can be used to extend the distance scale beyond where individual Cephids can be seen in the galaxy? All of these are very difficult problems. The best current value is now about 71 Km/s/Mpc both from traditional galaxy observations and from fitting the Cosmic Microwave background radiation fluctuations to a model. For more on this check out the entry on Hubble's Law at the Wikipedia.
3. Many scientists have contributed to the determination of better measurements, so it would be hard to name them all.
4. Probably it will change, since all measurements are prone to change as new techniques of measurement are devised.
5. it is still generally known as Hubble's constant. It has been known since 1916 and certainly the 1920s that in the theory which explains the phenomenon, namely General Relativity, that the constant is a function of time. So for this moment in cosmic time it is constant. However, theoretically it's value should vary over time.
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