The answer to the first question is no. The New Testament doesn't even give an example of anything Jesus made.

The answer to the second question is that it's possibly a metaphor (I'll explain how further down), but that it's perfectly plausible as history. The Greek word tekton doesn't just refer to carpenters, but basically referred to anyone who worked with wood or stone and commonly denoted anyone was involved with building projects (or ship building projects) in general. According to many scholars (e.g, J.D. Crossan) this was a low level, subsistence type job - basically they were construction workers and day laborers. This would fit in perfectly well with Jesus' social class and there is no particular reason to doubt it.

Moreover, there was a major building project going on about 4 miles (6.5 km) from Nazareth while Jesus was growing up at the reconstruction of Sepphoris. It's entirely possible that the males in Jesus' family made a living working in Sepphoris. This is speculative, but if they were tektons, they were tektons living an hour's walk away from plentiful work while the tiny village of Nazareth (estimates from archaeological surveys are that it might have been no more than 20 or 30 houses with maybe 100 people) probably would not have provided enough to support them.

As to the possible metaphorical meaning, there are a couple of uses in the Talmud of the word "carpenter" (in Aramaic) to mean a clever exegete of scripture - someone who can deconstruct and explain difficult passages. I'm going from memory, but from what I remember there is Rabbinic quote along the lines of "It's going to take a carpenter to figure this out." I think this meaning is unlikely, though. That's not how it's implied in Mark's Gospel.

I think there is at least some possibility that something in the ruins of Sepphoris could have been worked on by Jesus, but we'd have no way of telling what it was, of course (unless maybe he left some graffiti saying something like "Yeshua of Nazareth built this wall." What a trip that would be).