On the Sith Apprentice taking an apprentice
There doesn't seem to be a pre-requisite for the Sith Apprentice. The Apprentice may or may not take an apprentice of his or her own. If the Apprentice fails to defeat the Master and is killed, the Master will choose a new apprentice:
The Master will train his apprentice in the ways of the Sith, until one day she must challenge him. If she proves unworthy, the Master will destroy her and choose a new apprentice. If she proves the stronger, the Master will fall and she will become the new Dark Lord of the Sith, and choose an apprentice of her own.”
Dynasty of Evil p. 153
Since the Master can choose a new apprentice it doesn't seem to matter whether the Sith Apprentice had an apprentice of his/her own already.
Another quote explains what happens if the Sith Apprentice does have his/her own apprentice and the Master defeats the Apprentice:
“The Rule of Two dictates there can only ever be two Sith, so how can you take me as an apprentice if your Master is still alive?”
“If you accept my offer, you will accompany me as I go to face my Master,” Zannah explained. “But you must not interfere. If he falls, then I will take you on as my apprentice.”
“What happens to me if you fail?” Set wondered.
“If I die, my Master will need a new apprentice. If he judges you worthy, then you will replace me. If not …”
Dynasty of Evil p. 155
Again, the Master may choose the new Apprentice. That Apprentice may or may not be the failed Apprentice's apprentice.
The requirement for challenging the Master, which is hinted at in the previous quote, is that it must be a battle between the Master and the Apprentice at their full power and without any interference from anyone else (such as the Apprentice's apprentice):
An apprentice had to earn the title of Dark Lord, wresting it from the Master in a confrontation that pushed them both to the edge of their abilities. If Zannah intended to challenge him only after he was crippled by illness and infirmity, then she was unfit to be his heir. Yet Bane was not willing to initiate their confrontation himself. If he fell, the Sith would be ruled by a Master who did not accept or understand the key principle upon which the new Order had been founded. If he was victorious, he would be left without an apprentice, and his failing body would give out long before he could find and properly train another.
Dynasty of Evil p. 11
You remembered correctly that there could be no interference between the Master and Apprentice by a third party, but there does not seem to be a requirement that the Apprentice already have an apprentice. Perhaps the extra requirement you were remembering is that the Apprentice has to attack the Master at his full power, and that the Apprentice must initiate the battle.
On the possible extinction of the Sith
I've looked through the Bane trilogy and Darth Plagueis (the books which have the most thorough discussion of the Rule of Two), but I can't find anything specifically addressing the possible extinction of the Sith while there is only one (because the Sith Apprentice has just assumed the position of the Master).
The closest I can find is worry on the part of Zannah that Bane would die prematurely (when he was near death from poisoning):
If he died now, everything he had put into place over the last ten years died with him. Zannah would have to start at the beginning. She would have to find and train an apprentice, even as she was still learning the full extent of her own abilities. She would be stumbling blindly forward, beset by enemies on all sides. It was almost impossible to imagine she wouldn’t make a mistake that would lead to her downfall … and the extinction of the Sith.
She couldn’t allow that to happen. For the sake of their order, she had to keep him alive.
Rule of Two p. 291
This would be a serious setback to the Sith, especially if Zannah died as well. However, it probably wouldn't really result in the extinction of the Sith because the Sith are an idea, and ideas can't be killed. Sooner or later, someone would stumble across a Sith holocron, turn to the dark side, and re-create the Sith Order.
It's probably worth mentioning that the Sith weren't safe from extinction when there were many Sith Lords, either. Arguably, with many Sith Lords the Sith were in even more danger from extinction, both from the Jedi and internal power struggles. Recall why Bane created the Rule of Two:
The Sith had existed in one form or another for thousands of years. Throughout their existence they had waged an endless war against the Jedi … and one another. Time and time again the followers of the dark side had been thwarted by their own rivalries and internal power struggles.
A common theme resonated across the long history of the Sith Order. Any great leader would inevitably be overthrown by an alliance of his or her followers. Lacking a strong leader the lesser Sith would quickly turn against one another, further weakening the Order.
Of all the Sith Masters, only Bane had understood the inescapable futility of this cycle.
Dynasty of Evil p. 9
Bane would argue that the Sith were weaker when there were many Sith Lords, and they would further weaken themselves with infighting. Indeed, it was Bane himself who nearly caused the extinction of the Sith by tricking the Brotherhood of Darkness into using the thought bomb.