Have you ever looked at the breakdown of who writes the Linux kernel? It's mostly maintained by corporations, notably companies like Red Hat and strangely enough even Microsoft.
Yes, among other reasons such as solving a problem for themselves or enjoyment of the process. The point is that financial motivation doesn't even explain all the production in the economy right now, so obviously removing the financial motive won't remove all production. And a basic income doesn't imply no financial motivation anyway, how many investment bankers and programmers quit their job yesterday because they already have enough savings to eat and live in a crappy sharehouse for the rest of their lives?
