I don't think it's really fair to say they were short lived. The major players that that dominated the whole region, e.g. the Gupta Empire, Mauryan Empire, Delhi Sultanate, etc, generally were able to last for a good few hundred years each, which I would say is pretty comparable to the lifespans of polities throughout the world (States that exist for 1000's of years under the same institutions is actually pretty rare outside of Europe, although the reason why is beyond my expertise). For example, the two "golden age" dynasties of China, The Han and the Tang, existed for 400 years and 300 years each respectively. Outside of that, India is usually populated by a myriad of smaller states which, quite like you see in China (e.g. Warring States Period, 18 Kingdoms Period, Romance of the Three Kingdoms etc.) results in states periodically disappearing and reappearing due to the constant warfare that kind of geo-political situation usually brings.

So basically, they were more stable than you're giving them credit for, and when they were unstable their relative fleetingness is pretty average.

You seem to not really have much context for the second question, so I'll try to give you a brief intro into how it happened.

For your second question, that's because the Greeks ruled their! After Alexander the Great's conquest of the Persian Empire, his top generals divided the Empire via the Wars of the Diadochi (the Diadochi was the collective name of the generals). The rule of what was roughly equivalent to the old Persian Empire minus Egypt was taken by a guy named Seleucus, who founded the Greek Seleucid Empire. The Seleucid Empire stole a page from the Persians and governed via the usage of Satrapies (basically vassal kings that were given relative autonomy in exchange for taxes and the providing of troops), and the farthest Eastern Provinces of the Empire, roughly equivalent to what is today Pakistan, was a satrapy. In 256, the Satrap, Diodotus, rebelled against the Seleucid Emperor Antiochus II and won independence. This founded what we now call today the Greco-Bactrian kingdom (Greco because Greek, Bactrian because that's where it was based, it's a region in what is today Iran and Pakistan). The various happenings of the kingdom for the next hundred years (fracturing, invasions, etc) caused the establishment of a few different Greek states, giving us all of those minor Greek kings you were referring to!