Which European capital is the oldest?

Very difficult to say. Depends on whether you mean oldest as a settlement, oldest as a capital of somewhere, or oldest as the capital of where it is currently the capital. For example, Rome is traditionally dated to 753 BCE (I think - haven't checked it), but it has only been the capital of Italy since the 1920s, as Italy didn't exist as a nation until 1860, and Turin was the capital most of the time. Athens has a very long history, but for most of it, Athens wasn't a capital. San Marino (city) has a fairly good claim, having been the capital of San Marino (republic) since 885 or thereabouts.

Very few capitals in Europe have enjoyed a completely uninterrupted spell between first becoming the capital and the present. For example, when did Winchester last function as the capital of England? Moroever, if one goes back far enough into the Middle Ages, the capital was often where the King and his court were.

I would say it is Athens because it was the capital of Greece from the start of the civilization and also it is now the capital of Greece. So I will go with Athens.

Softly, softly. Was Athens in fact the _political_ (as well as cultural) capital of Ancient Greece?

No. Ancient Greece consisted of many city states and other odd bits, who were very often at war with each other. They didn't even totally unite to fight the Persians. Sparta and Ithaca are two examples of states. And for many years Greece was part of the Roman Empire, and then the Byzantine Empire, and the Ottoman Empire. Athens only became a capital as such with the establishment of the kingdom under Otto (nothing to do with the Ottomans - just a coincidence and they booted him out again fairly soon). Paris and London have fairly long continuous histories as capitals, as well as San Marino, but most other modern European and Asian captals don't. Beijing became capital of a united China in 1368, but doesn't have a continuous history from then, as the capital has been at Nanjing at times. But the three European ones I've mentioned beat that anyway.

What about Reykyavik?

Reykjavik was first settled in around 870, but there wasn't an Icelandic nation as such until 930 when the Alþingi was first established. And Iceland was a dependency for quite a long time until home rule was granted in 1874.

Valletta, perhaps. It has been the capital of Malta for a good 2000 years.