It seems that logical fallacies are regularly committed in the course of daily political dialogue. For example, many politicians support their policy decisions through appeal to the emotions of the electorate rather than their faculties of reason. If philosophers possess the tools to dissect the logic and substance of political discourse, why don't philosophers play a greater role in public life? Why isn't their a panel of philosophers to transparently dissect and scrutinise the speeches, policies and actions of contemporary politicians?

I do not think that specialized philosophical training is needed to dissect the speeches of candidates for elective office. Only the rudiments of critical thinking, a willingness to follow good arguments wherever they lead, and some knowledge of the relevant issues is needed. In universities and colleges, philosophers teach classes that aim to cultivate the skills useful for analyzing arguments and recognizing their strength (or weakness). We do our part in that way -- by helping to produce an electorate that can think critically, reason carefully, and express itself clearly. But the rest is up to you!

I read recently a comment by a philosopher that Karl Popper's "falsifiability" theory is considered obsolete. Is this so? I always found it to be quite useful. If it's obsolete, what rendered it so, and by what was it replaced?

There are several considerations that count strongly against Popper's "falsifiability" criterion, but I'll mention just two. Remember that Popper's criterion is intended to distinguish science from non-science (or pseudoscience) on the grounds that a theory is scientific if and only if it is 'falsifiable', i.e., there are possible observations that would logically contradict it. Now for two quick arguments against this view: 1) Consider a statistical hypothesis, such as "This coin has a 50% chance of landing heads and a 50% chance of landing tails on any toss." Statistical hypotheses play a very important part in many important scientific theories (such quantum mechanics, evolutionary biology, statistical mechanics). But no possible sequence of coin toss outcomes logically contradicts this '50% chance' hypothesis. Some outcomes support the hypothesis; some disconfirm it. None falsifies it. 2) Many significant scientific hypothesis make no observable predictions all by themselves, but can...