It's 2. :)
... self-replicating organisms came into existence on this planet
and immediately began trying to get rid of each other, either by
spamming their environments with rough copies of themselves, or by
more direct means which hardly need to be belabored. Most of them
failed,... Like every other creature on the face of the earth,
[Godfrey Waterhouse IV] was, by birthright, a stupendous badass,
albeit in the somewhat narrow technical sense that he could trace
his ancestry back up a long line of slightly less highly evolved
stupendous badasses to that first self-replicating gizmo — which,
given the number and variety of its descendants, might justifiably
be described as the most stupendous badass of all time. Everyone
and everything that wasn't a stupendous badass was dead.
As nightmarishly lethal, memetically programmed death-machines
went, [his parents] were the nicest you could ever hope to meet....
The parallel that comes to mind is the mitochondria within animal cells, they probably originated as independent bacteria.
That is, DNA evolved twice on Earth. That seems like "life beginning more than once" or something very close.
Note that almost everyone involved thinks that mitochondria and chloroplasts are prokaryotes captured one way or another by Eukaryotes, so the distinction about two evolutions of DNA gets a bit messy.
As to your questions, they have vexed scientists and philosophers for ages, so there're no easy answers, but I am assuming you didn't expect one!
The conditions for life as we know it are bounded, so that reduces the surface area of possible places life could exist. Add to that, the requirement that the bounds not change dramatically for long periods of time, and they reduce further.
The universe is pretty huge, though, so there's a lot of opportunity for life out there, but because it's so huge, and because life is so fleeting, there's a lot less opportunity for two unique forms of life from different planets, solar systems, or galaxies to actually cross paths at the same time, or even to do so before one dies off.
For further reading, check out the Drake Equation  and Fermi's Paradox .
Perhaps you meant to ask if there could have been no universal ancestor? In that case I think most evidence points the other way. There could possibly have been multiple 'origins' of life but those seem to have died out or have been driven to extinction by our ancestors.
Firstly, the current environment is radically different from the conditions in early earth:
* Atmosphere composition has changed a lot, and largely due to the presence of life (more oxygen, no more ammonia, etc).
* In early earth it would be possible for complex organic compounds to accumulate in the environment. Nowadays, those would be rapidly consumed by life forms.
There is also a matter of competition:
* Almost every ecological niche on earth is already occupied by life and there is very little space left for alternate life forms to evolve into.
* Current life forms compete against each other and that competition might also affect other nascent life forms. For example, one of the leading theory on the origin of life predicts that early life was largely RNA-based but nowadays lots of the planet is coated in RNA-digesting enzymes that organisms excrete as a form of protection against viruses. In fact, scientists working with RNA have to be very careful to avoid having their RNA be degraded by environmental enzymes.
How can anyone make this claim? Is there a single example of a living organisms that is not DNA based? If everyone organism alive today uses DNA then it is reasonable to assume that the LUCA was DNA based (which does NOT preclude the possibility that life started off as RNA based). I guess that's wikipedia for you.
These are from the New Scientist article that wikipedia referenced. I don't understand the references here, but hopefully someone will see this that can help explain exactly how an organism doesn't have DNA, only RNA.
There's a long-standing debate over whether enzymes or nucleic acids came first. In modern cells, enzymes are necessary to catalyze energy-producing reactions and other chemical reactions, and nucleic acids are necessary to store the information necessary for self-replication. It seems unlikely that nucleic acids and enzymes both were randomly spontaneously generated and worked together in the first self-replicating chemical reactions. Instead, most people have theorized that either early cells used proteins to store information or used nucleic acids to catalyze reactions. It has been shown in the lab that some reactions can be catalyzed by RNA.
So, the theory goes something like (1) self-replicating RNA spontaneously arises that both catalyzes reactions and carries information (2) mutations in some of the RNA causes it to create enzymes, which make much better catalysts and out-compete RNA-only life (3) RNA-enzyme life mutates and starts using DNA as a more stable storage format than RNA, making those DNA-RNA-enzyme organisms more robust and these out-competed all RNA-enzyme organisms.
Consider that cells are mostly ribosomal RNA/units by mass.