Why did abiogenesis only happen once?
It may never have happened again, however, since all lifeforms on Earth today are similar on a molecular level (DNA), suggesting a common origin.


An important distinction to make here is that all extant life on Earth has a common origin.

It's completely possible that abiogenesis occurred many times, but whatever organisms emerged as a result became extinct early on or are not preserved in the fossil record.


  Immediately after the first spontaneous abiogenesis, environmental conditions on Earth changed dramatically, making a repeat impossible.


Nope. The red bands and subsequent dating techniques suggest it took a very long time for a change to occur.


  The first organisms that arose consumed any subsequent organisms...


Possible, but we have solid data suggesting the first functional organisms were autotrophs. That is, they made their own food from whatever they were exposed to, and the predator/prey relationship had yet to play a major role.


  The common origin theory is in fact false, despite the similarities between organisms, and abiogenesis did occur multiple times in the same way.


The Common Origin Theory - as you've defined it - is almost certainly true. DNA is a constant, as is RNA, mitochondria, etc. 

That, however, doesn't mean abiogenesis couldn't have happened more than once. As I said above, the Common Origin Theory applies to extant (living) species, not necessarily all organisms that have ever existed. Species go extinct all the time.


  Spontaneous abiogenesis never occurred on Earth after all because the conditions never allowed it; instead, a proto-organism arrived on Earth from a planet where the conditions do (Panspermia).


That merely pushes the question of abiogensis back a bit. Why did it evolve on another world and not Earth, then? How did it survive in space? Why was it so well suited to Earth's environment if Earth-like planets are (relatively speaking) rare? 


  The standard theory of geological history is wrong (i.e. the Earth was in a "fertile" state for much longer before the eventual origin of life than commonly thought, due to some unknown mechanism causing radiometric dating to give wrong results).


Estimations and conclusions have come from more than radiometric dating. We have fossil records, geological records spanning continents, and even ice-core records from the poles that both confirm radiometric dating and establish themselves as independent evidence.


  None of these seem likely to me, yet even less likely I find the idea that life arose spontaneously almost immediately (within a few hundred million years, possibly even faster) after the Earth's crust solidified, and never again afterwards.


Why?


  What are the currently held theories on that matter? Both random speculations and references to relevant publications are welcome.


This SE is not the place for random speculations. I don't have publications handy, so I may return and edit my answer when I have the impetus.

The most widely held theory is that all extant life, with a few possible exceptions, has evolved from a single organism that existed billions of years ago. That organism may or may not have been the only one to arise. If it was one among many variations of progenitors, we have yet to find the progenitors in the fossil or biological records. Given the extremely long time span between now and the beginnings of life, we don't expect fossil records to exist of all the organisms that existed some 3 B.Y.A. so the best we can do is speculate on what might have existed to account for what we see today.