Yo dude, thanks for the link. I didn't know it was World Philosophy Day. Your BBC article covers four classic questions, each of which I've written about before.
1. This is called the Trolley Problem, and I use it on fellow libertarians a lot. The crucial consideration is how much freedom you have in choosing who is the one who will be sacrificed to save the many. If circumstances (or a bad guy) picks the one, then the right answer should be clear. Otherwise, you need to set up a lottery, and you need to weight things according to expected lifespans, objective quality of life, impact of the losses on others, risk of setting precedents, etc. Luckily, these tragic "lifeboat" scenarios pretty much never happen, and that is why we're not used to making the hard choices involved in them. The choices would be emotionally hard, but they're not philosophically paradoxical.
2. This is called the problem of Theseus' Ship. The answer I give in my book is: "A given entity is identified through time with its closest close-enough continuous-enough continuer. A continuer is an entity which is similar to a previous entity and exists because of it. A continuer is close enough if it retains enough of the original entity's properties. A continuer is closest if it retains more of the original entity's properties than any other continuer. A continuer is continuous enough if there is no extraordinary discontinuity in its relationship to the original entity." This whole topic of identity (including forked and joined identities) is covered in one of the best philosophy books I've ever read: The Metaphysics of Star Trek. If I haven't bought you a copy before, then you're getting one for Xmas.
3. Yep, there is no absolutely certain synthetic (i.e. empirical) knowledge. We've known this since Hume. As I say in my book: "All synthetic propositions (including this one) can only be known from experience and are subject to doubt." The crucial thing is to understand the level of confidence to assign to synthetic propositions, and to understand the ways in which they might be false.
4. The problem with free will is that people think of their mind/soul as something apart from the universe, rather than as a subset of the universe. I write: "Free will is either of the doctrines that human choices are a) determined internally rather than externally (volitional free will) or b) not pre-determined at all (indeterminate free will). Determinism is incompatible with indeterminate free will, but is compatible with volitional free will if agents have internal state that influences (and thus helps determines) their actions."
These are great classic problems. Another really good mind-twister related to free will is http://en.wikipedia.org/wiki/Newcomb%27s_paradox. Infinity is also a great mind-bender, such as the way it lurks in http://en.wikipedia.org/wiki/St._Petersburg_paradox. I bet you would like http://en.wikipedia.org/wiki/Doomsday_argument.
Yes, I've tivo'd Parallel Universes, can't wait to watch it. I'm a big fan of modal realism -- the theory that possible universes are just as "real" as this one. It's related to the biggest of all philosophy questions: why is there something instead of nothing? My answer: "A merely possible universe would be perceived by its merely possible inhabitants no differently than our actual universe is perceived by its actual inhabitants. [Modal Realism says "actual" just means "in this universe", and so is redundant when talking about our universe.] Thus, our universe might merely be the undreamed possible dream of no particular dreamer."