I couldn't find anything on calculating half-lives (too many responses about using them in decay calculations). We'll need one of the physics gurus ...
Calculating Half-Lives basically involves calculating decay rates. Calculating decay rates is a very difficult, technical and involved subject. I was going to give a post describing it, but then I realised that the post would have to be five posts long to really convey anything properly.
Basically you've to right down the Lagrangian, the entity that describes the particle interactions, then you've to calculate in a rest frame that simplifies the calulcation. Then you've to compute the Feynman Diagrams......blah,blah......... and there's a huge amount of dirac algebra, unless your dealing with simple scalar particles with simplified Lagrangians. Then after you've done all that you'd still have only calculated one mode (i.e. only one of many possible decays), any given particle can decay into quite a different number of things.
Quantum Mechanics isn't really enough to deal with decay rates (in fact it treats them artificially, at the very least I don't like it), you need Quantum Field Theory.
Even then I've only ever truly worked out free-particle decays. Actually working out decay rates for bound states like atoms would require teams of people (that's what 90% of the physicists at the Manhattan Project were doing) or a computer.
Yes, I think it has been accurately done for at least thirty years.
And what kind of agreement with measurements do you get?
As perfect as possible. It is not within our technological abilities to detect the difference between the predicted value and the observed value. If there is a difference it would be a while before we have the technology to detect it.
What are the inputs? E.g., speed of light, h bar etc.
h bar and the speed of light aren't inputs, only conversion factors. The true, non-chosen, inputs would be the fine structure constant, the weak and strong coupling constants and a few particle masses.