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1. I don't know how it is being measured in this case but yes, periods of time much shorter than a picosecond (1.0x10^-12) can be measured. A picosecond is about how long it takes light to go one millimeter.
2. Einsteins theory is not that it would take infinite energy to excelerate mass to the speed of light. The relationship between mass and the energy of such acceleration is an implication of his theory but not the whole thing. The speed of energy (light) is very much at the core of Relativity and if it has been passed this would have huge repercussions. First of all, for energy, "acceleration" is pretty much a meaningless concept as it only has the one speed (in a vacuum; the speed of light is different in other mediums).
3. So far as I know, no the experiment hasn't been repeated. Though if the newspaper story is an accurate reflection of the claimed science (certainly no guarantee, newspaper generally suck at science reporting) then it will quickly be tried all over the place. That said, there are already quite a few known phenomena (such as the quantum tunneling mentioned in the article) that appear to violate relativity at first glance but actually don't. If it is in that class it may be more hype than substance.
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