Yes, they do. But there is such a bewildering array of hybrids from different species floating about now that the only real way to be sure you have viable seed is to plant it and wait :)

I would say some species do and some dont. The bedding type do I believe. All the native annual species do and will set seed when very small. Many of the Asian annuals are as well. Impatiens omeiana isnt at least I have never seen a seedling the same with arguta though I have seen pods but never have I collected seeds of it.