First one, is what name is this Rex begonia?
Second question is, what is this 'bud/knub on the leaf and is this unusual? I have never seen one before.
Well, I don't know the name of the rex but the bud/knob on top of the leaf is the formation of new plantlets (at least that is what most of mine do when I propagate from a leaf cutting). Typically you see them after the leaf is severed and rooted. Maybe with your climate it doesn't matter.
When I propagate a leaf, I get a 'miniature plant, not a 'flower/knub' like this.
The 'bud' is prickley, not soft and doesn't resemble the small bloom that is produced.This photo shows it from the side.
The small 'bloom'
Your photo did not come through.
Here is what I see mine do if it helps you any.
These were all started in water.
Taliensis which was from last winter. I potted the leaf up after roots formed in water. It is showing little leaflets in this picture.
'Purple Curl' which I started 4 weeks ago.
'Emerald Star' just starting the "budding/knobbing" for lack of a better name for the bumps.
I don't know the name of this beonia (but it is a beauty when full grown) showing it has got some good size plantlets after several weeks.
Finally 'Peace' which has developed leaves after 6 to 7 weeks.
Oh, how interesting! When I rooted begonia leaves, plantlets would come up from the soil, from the end of the petiole. And in your pictures, hcmcdole, you have them come up from the base of the leaf itself! How cool! Is this something particular to rexes? I only ever rooted rhizomatous begonias... Do you do anything to the base of the leaf (say, do you scratch it, or something like that?) to get the plantlets to grow from there? Do you set the leaf so that its base is touching the soil/water? This is very interesting, thanks for posting the pictures!
Rexes are more likely to sprout from the petiole/blade union than other rhizomatous begonias but the first picture was of B. taliensis which is not a rex. Rexes will also sprout from the soil as well as the center of the leaf.
Here is another taliensis with budding on the leaf and sprouts coming up in the soil.
Here is B. 'Comtesse de Montesquieu' exhibiting the same behavior as most rexes. Note the leaflet coming up from the soil on the bottom edge of the mother leaf. All the other leaflets are at the union.
Finally here is B. 'Emerald Star'. Is it a rex or something else? The leaves are similar to B. 'Benitochiba'.
I should mention that B. soli-mutata does exactly the same thing - buds at the leaf union and from the soil.
Thanks for the explanation and more pics! This is very cool :)