Starting from the bottom:
We apologize if some of our questions seem too general or a bit naive, though we believe that answers should exist which are comprehensible for non-expert physicists.
The authors are science journalists. If you have "naive questions" to which you believe general-public-accessible "answers should exist", you have any number of productive ways to address that rather than attempting to upload them to the ArXiV.
What were the predictions? Our impression was that the Higgs signal was searched for in every energy region and eventually found in the only one not excluded before. Was there any theoretical prediction of its mass earlier than 2011?
There was an experimental constraint requiring a standard-model Higgs to be on the light end of the scale (closer to 114 than to 200 GeV, say), but a SUSY Higgs could have been heavier. There was no specific mass prediction, which is why you have to search such a broad region.
Two photons. So what? The first and most important evidence is an excess of photon pairs. Virtually every particle-antiparticle pair created by the collisions decays into two photons. How can one read from this any characteristic of such a peculiar and unique process the Higgs mechanism is claimed to be?
This is simply untrue. First, particle-antiparticle pairs created at LHC energy typically fly in opposite directions into your detectors. The familiar "fact" about antimatter, that it meets matter and annihilates into photons, is familiar only from positrons at low energies. Nonetheless, there are lots of two-photon events at the LHC, but these occur at random energies with a characteristic broad distribution. The Higgs signal (just one of several complementary signals) is a photon pair
at 125 GeV, which is an extremely specific signature of a 125 GeV boson.
Is this a triumph of the standard model? It seems that at least some branching ratios (decay probabilities to various particles) do not match the expected values,
You're talking about, like, 0.5-sigma fluctuations. This is equivalent to flipping a coin 10 times, getting 7 heads and 3 tails, and complaining that you were told to expect 50/50 results.
How is radiation damping controlled? A high-energy collision of protons, as for all charged particles, implies that enormous (negative) accelerations must occur. Accelerated charges necessarily loose energy due to radiated photons. However, a complete theory at what wavelength, rate and direction these photons are emitted by accelerated charges, just doesn’t exist.
They built an accelerator and it works. The protons go round and round the way the accelerator engineers told them to. If you don't think the protons are going around the accelerator, or if you don't think we know enough E&M to accomplish this---notice the difference between "enough E&M" and "all E&M up to the Planck scale"---then your problem is not with the Higgs discovery but rather with all modern engineering. Did you know that your microwave oven also works by accelerating electrons? AT LEAST THAT'S WHAT THEY TELL US.

How do you remove a background of one trillion pairs?
Using computers.
Is it reasonable to claim that 600 million collisions per second can be modelled to such an incredible accuracy?
Yes.
Is the lifetime of the Higgs irrelevant? The standard model predicts the lifetime of the Higgs to be 1.56 · 10−22s. However, the photon excess at 125 GeV has an approximate width of about 8 GeV.
The LHCs have detectors with finite resolution;
everything looks at least about 8 GeV wide when you measure it with calorimeters. (It's like: if your car's speedometer reports that you are driving at 65 +/- 1 mph, does this tell you anything about the quantum-mechanical uncertainty in the position of your car? No, it just tells you that your speedometer has a 1mph error bar. If you want a direct measurement of the Higgs width, please write your Congressman and ask for funding for the ILC, an electron-positron collider.
Is this an explanation of masses? It is often stated that the discovery of the Higgs boson would explain an old riddle of masses. Indeed, Einstein, Dirac and Feynman wondered why the proton is 1836 times heavier than the electron. If the answer should be ‘because it couples 1836 times stronger to the Higgs field’, where precisely lies the epistomological insight?
The "riddle of mass" is not "why is the top mass 173 GeV and the muon 0.1 GeV and the tau mass 1.8 GeV and ... ". The riddle of mass is "how can these masses be anything other than zero"? The Higgs mechanism answers the latter question. It does not answer the former.
How many numbers are in the game?
Experimentalists use standard modern statistics tools, and awareness that the number-of-constraints must be smaller than the number of degrees-of-freedom is ... well, it's extremely elementary.
What are the model-independent results? If the results of the LHC should confirm the standard model, it seems to be a fair idea not presupposing the validity of the standard model.
Three clicks on the CMS web page got me to a list of 53 papers in the category "Standard Model Physics".
Why not public data? The scientific method relies on reproducibiliy of results. We appreciate that the instrument experts apply their corrections and calibrations to the raw data. But we miss a publicly available dataset that simply says how much energy, at a given time and location in a specified detector, was deposited. Why does an experiment with huge resources not provide public access to such model-independent results?
First: the LHC dataset is one of the largest datasets ever amassed on Planet Earth; storing it and making it available to
the scientists was only barely possible.
Second: the events are more complex than you think they are. You want a list of "what energy was deposited in what detector at what time" and you think you can get physics out of it? This is like going to the Hubble Space Telescope, asking "what voltages were on what wires at what times", and thinking you can use this to cross-check Hubble's dark-energy data.
Third: if you have an idea for an analysis you want to do, write up your idea. Figure out what data you would like to plot (and with what cuts/triggers); figure out what the Standard Model predicts for that plot (including uncertainties); if you have non-SM ideas, figure out what those ideas predict too, and whether the differences are detectable. Write it up, post to hep-ph, talk to some experimentalists, and see if anyone wants to do it. This happens all the time.