Is it better to take a half dose of paracetamol and a half dose of ibuprofen together rather than a full dose of either?
There are many different reasons for an acute pain and inflammation (both directly via pain receptors, and indirectly via swelling etc.) plays not the least role in this phenomenon. Therefore the combination of a classic analgesic without anti-inflammatory activity (as paracetamol) with a strong anti-inflammatory agent (as ibuprofen) is a reasonable one and potentially alleviates the pain manifestations in more cases than a single administration of every agent.

You are right with you suggestion that the doubled dose could be dangerous: even though the complications of Paracetamol administrations are not often, they are very dramatic (acute liver failure) with rather pessimistic prognosis. This is why one generally tries to half the doses if combined with other similar-acting agents (historically paracetamol was for the long time a member of the non-steroid anti-inflammatory agents).

Caffeine is generally believed to be a pain reliever, especially as concerns the pains with migraine background (source). One review, however, points out that there are still not enough data to support this belief (source).