If I prune the flowering stalk of the Echeveria or its relatives just under the first flower, leaving a stem with the leaves on it, will it branch? make new flowering stalks? New plantlets which I can root?
If you leave the flower stalk attached to the plant, it will slowly wither and die, and eventually fall off. Letting the plant go dormant after flowering, by stopping all watering for a few weeks, will speed up that natural process.
Cutting off just the flower head is rather pointless, because all you have done is create a wound where pests and diseases can attack the plant more easily.
The leaves on the flower stalk can be used for rooted cuttings just like the other leaves, but since they are usually small they are more likely to dry out and die before they develop roots than cuttings using bigger leaves.
If you really want to propagate a lot of these types of succulent, just "disassemble" a complete plant and use every leaf as a root cutting!