"Who exactly is able to become a Kishin?"
It has been stated in both the anime and the manga that humans cannot consume souls, only demon weapons. Yet Don Al Capone, a human, was seen eating human souls in the anime. The result of his soul consumption turned him into a Kishin egg, the first step in becoming a Kishin (to my understanding, first you become a Kishin egg, then a Demon Sword, then a Kishin).
However, that only applies in the anime. In the manga, there are no "Kishin eggs" only Shinigami's list of evil human souls that only Lord Death can distinguish to be evil.
If you are referring to the manga, I'd say only demon weapons can become Kishin except for Asura who consumed his weapon which was already a Demon Sword. By swallowing his weapon he gained the ability to eat souls.
However, in the anime humans can become Kishin eggs, but it seems unlikely that they would turn into "Demon Swords" because they are still humans--not Demon Weapons. On the contrary, Jack the Ripper's hands appear to be completely replaced with claws which, in some way, makes his body a weapon, so I suppose it's plausible.
"What would've happened if Asura's Demon Sword continued to eat human souls?"
If Asura continued to feed Vajra human souls, then Vajra would have been the one to become the Kishin, so long as he wasn't eaten first.
"Could a Demon Weapon become a Kishin without a meister?"
Demon Weapons are fully capable of consuming souls. And, assuming that they are able to kill humans on their own, yes, they could become a Kishin without a meister.
However, some weapons such as Nygus or Liz (or Patty but not together as a Weapon/Meister pair, individually) are probably unable to turn a specific part of their body into their weapon form that can be used effectively like Soul and Justin Law. Such people may be incapable of becoming a Kishin solo.
