What isotope has the shortest half life?
The question is ill-posed.
To begin it should be "*What isotope has the..." and even then the answer is "We don't really know, as there are some we have produced too few times to have an accurate measure for but they sure don't live long." Looking at the low-mass end of the periodic table I find some described in terms of the linewidth---which means really short.
Isotopes with halflives measured in 10s of seconds have been put to use from time to time. And the shortlived low-mass isotopes are useful in calibrating underground neutrino detectors.
I agree with dmckee in that there are isotopes with extremely short half-lives for which we do not have sufficient numbers to produce a reliable value of half-life. Additionally, for an arbitrarily small half-life, one could imagine an isotope that would theoretically have a half-life less than that (depending on which model you prefer); once you get a half-life below a second or so, the material is essentially useless anyway and it's just an academic question, except potentially in some astrophysics applications.
As to how to redefine time, there have been talks about that for a very long time. During the French revolution they even made clocks representing a 10 hour day with 100 minutes per hour and 100 seconds per minute. Honestly, I kind of like that setup, but it really is quite arbitrary and it is highly unlikely that any alternative time system than the current metric one will be adopted within our lifetime.