On the Origin of Life
Observation One: The chemical elements which make up the bulk of all life; of all living things, just happens to be among the most common and abundant chemical elements in the Universe - CHON (Carbon; Hydrogen; Oxygen and Nitrogen). That life is basically composed out of the same very common stuff as the Universe itself can't be a coincidence. Life is NOT made up or out of exotic stuff like gold, silver, bromine, uranium, mercury, arsenic, etc.
Observation Two: Our galaxy is roughly ten billion years old. That's a lot of time for chemical reactions to happen in; for chemistry to strut its stuff from inorganic chemistry to organic chemistry to biochemistry.
Observation Three: There is a vast amount of real estate out there even in our own galaxy alone. One can even include interstellar space itself as part of that real estate since lots of organic chemistry / organic compounds have been observed in the depths of interstellar space as well as in meteorites and in comets which can bring organics to Earth (or any other environmentally friendly location).
Observation Four: There is a vast, vast range of environmental conditions within our galaxy. In fact, just think of the massive range of environmental conditions present within our own solar system.
Observation Five: Therefore, given the commonality of chemical elements between life and the rest of the Universe; extremely lengthy time periods; a wide variation in environmental conditions throughout, it's probably inevitable that just-so Goldilocks conditions giving rise to life would arise at least once. So there's no reason why Earth couldn't have been Ground Zero for life's natural origin.
Observation Six: It just takes one origin of life event to take hold for then that life can evolve and adapt and actually spread throughout the galaxy - there's more than enough time - via a concept called Panspermia.
There's actually some degree of evidence for Panspermia. First off, once the newly formed Planet Earth settled down, life arose extremely quickly, perhaps too quickly for a terrestrial in-house origin. Secondly, there's that Mars rock discovered on the Antarctic ice cap which contained about four separate and apart biological indicators. Lastly, there are microbes on Earth which even inhabit the insides of nuclear reactors and obviously can withstand massive doses of radiation; probably the last survivors after a nuclear war. There's no rhyme or reason for that to have evolved if these critters had a totally terrestrial origin and upbringing. But in space where radiation exists in abundance, well that's a different story.
I wouldn't call it a coincidence that life is carbon based, it can make stable connections and is still quite reactive to form new connections. The other elements are just less suited for life (under those conditions that we consider normal). Also you have to keep in mind that heavier atoms only form either a) after a long time or b) in super-nova explosions. So those newer elements where "added" to the universe only after enough Stars exploded.
There is as far as I know some evidence for life forming elsewhere, but it is not out of the question that this evidence formed differently.
You say there is no rhyme or reason for those critters to withstand high radiation. I see no reason to conclude that. There are many life-forms that survive extreme conditions and it's perfectly plausible that they formed on Earth.
But I think for the question of the Origin of life it doesn't really matter where it started. If the panspermia theory is correct it would still be more interesting to know how that happened. I personally believe (terrestrial) life evolved on Earth and that there are other places in the universe, where we could find life. On earth it happened so fast that I would really be shocked if there was not other life out there. Panspermia is plausible but I would prefer if it was not the case for then we would have to find the planet where "our" life started and that is a question that we might never be able to answer depending on how far away it started and how widespread it is. So we would never know our true origin.
For your first point. Mutations can be beneficial, detrimental or neutral for the survival of an organism or species. So this resistance to radiation could be such a neutral mutation that had no benefit or disadvantage to that organism. But I think it is more of a byproduct to having to withstand other extreme conditions like high pressures and temperatures. Things that do exist on earth.
I don't like the explaination of directed panspermia because it tries to explain the development of simple organisms to complex organisms by presupposing complex organisms, which leads us pretty much to the beginning again. Furthermore those complex organisms must be more complex or at least more advanced than us for achieving this. But we have no evidence of more complex life than us right now. Of course this might change, but I'll keep waiting for the evidence to come until I make such assumptions.