I did some research on this specific topic when it was first mentioned to me, and the best answer is 'no one really knows'.

Burke's Peerage stated this back in 1986, and they claim that the connection point is Zaida of Seville. Finding sources on all this is next to impossible, because there are conflicting and incomplete sources.

I'll do my best to explain the general idea of Zaida, and why this question is impossible to answer.

Zaida was a Muslim princess. That much is about the only thing that we can say with certainty. Some sources say she's the daughter-in-law of Al Mutamid (the king of Seville at the time), in which case the connection doesn't work. Other source say that she's actually his daughter. When Seville fell, she fled and took refuge with Alfonso VI.

Some sources say that she was his mistress, who then converted to Christianity and renamed herself 'Isabel'. Only other sources indicate that Alfonso's fourth wife was Isabel--but they don't make the distinction of if it's the same Isabel or not. So she might have been just his mistress, or she might have married him and been his queen. If she was just the mistress, she has just one child, who died early, while if she was also the queen then she had three.

Basically, in order for Queen Elizabeth to be descended from Muhammed, Zaida must be the daughter (not stepdaughter) of Al-Mutamid, and she needs to be the Queen Isabel who gave birth to Alfonso VI's children.

There's really no evidence either way, but the whole connection requires a giant grain of salt, since it requires a whole lot of things to be true of which there is no proof.

As to your second question, I unfortunately have no idea.

One of the few digitized sources found online is this one. Page three talks about Zaida, and exactly how confusing the record of her life is.

I'll use this thread for my question, can we trace back the house of Windsor to the house of Tudor, or even earlier?