Why do we assume that particles and their antiparticles have the same half-life? Has it been proven experimentally? What would the consequences be if their half-lives were different?
 A: Particles and their antiparticles having the same half life is related to the C symmetry (charge symmetry), which roughly states that processes for particles and antiparticles (that is, if you have a system and you apply the C operator on) have identical probabilities.
It is not true that all of them do, though, as the weak interaction breaks C symmetry, and more generally the CP symmetry (both charge symmetry and parity symmetry). You can check various mesons and hadrons for what is called the decay-rate asymmetries, defined as 
\begin{equation}
\mathcal{A} = \frac{\Gamma_- - \Gamma_+}{\Gamma_- + \Gamma_+}
\end{equation}
$\Gamma_-$ being the decay rate and $\Gamma_+$ the decay rate for the process after the CP transformation is applied. This has been observed to not be 0 in some baryon decay. 
A: A small point in addition to Slereah's solution: it is only CP symmetry that is required to have the same half life for particles and antiparticles.
A brief story of fundamental symmetries: for a long time it was thought that the laws of physics behave the same in a mirror situation (parity P), and also that antiparticles are exactly indistinguishable from particles (C symmetry). It was discovered in 1957 by Wu et. al. that parity symmetry is indeed violated, to great shock. Later it was also found that C is violated. For a while it was assumed that the combined symmetry CP holds (i.e. the physics looks the same when you take a mirror image and replace all particles with antiparticles), but in 1964 it was revealed that CP is violated (but only very slightly). Measuring the exact amount of CP violation is a hot topic in fundamental physics to this date - in particluar with the "Strong CP problem": there is no good theoretical reason for the strong force not to violate CP, so why doesn't it?
Now, the assumption is that the combined symmetry CPT (CP combined with time reversal) holds, but we have very good theoretical reasons to believe this to be true. Still, tests are carried out to test this.
