1) Whenever you finish a project the next one pops up right away. Is this completely random or does something influence it? Like what zone you're in, the last artifact you collected, or something.

Completely random. The only thing that influences what you get is your skill, but only indirectly. Your skill determines which items are available to be chosen, but the choice is still random.

2) If you are working on a project that needs for example 32 artifacts, what happens if you continue to pick up artifacts when the bar says 32/32? Are those artifacts lost or are they stored and used on your project? If it's the latter, is it then shown anywhere how many artifacts you have extra or is that 'hidden'?

Artifacts are never lost. If you are 32/32 and pick up more artifacts, you will show as 35/32 and start the next one with 3/32.

3) I see people talking about buying artifacts, is that possible? I don't see them anywhere as an object so I wonder how to sell them.

You can buy key stones, which are items specific to each race which are equivalent to 12 fragments. These drop randomly when you are digging. These are things like Highborn Scroll, Troll Tablet, etc.

It’s also possible that you’re referring to the final output. Once you click the Solve, button you will receive an actual item. Common items (greys) can be safely sold to a vendor. Blues can’t be sold- so if you no longer want it, I guess you’d have to destroy it. Epics (at least the gear) can be sold, but the prices aren’t that great, so you’re probably better off keeping it. All of the non-common output from this profession are BOP, so they can’t be bought/sold on the AH.