I happened to see this TED talk by Lesley Hazleton. She is an agnostic Jew and without bias she explains that Quran doesn't speak of 72 virgins. It is simply a misguided verse of the Quran. The verse no 33 of Sūratl-naba. I am a bit perplexed here.
Is that a misinterpretation of the Quran?
I can't give you as erudite an answer as the others, but I can address your two specific questions about the Qur'an. My main reason for offering this answer is that a significant portion of those who watch TED talks are non-Muslim Americans who have almost zero familiarity with Islam, and the familiarity they have is usually based on wild propaganda.
So in case you (or anyone else reading your question) find it useful, here is the perspective of a non-Muslim American who has read, from begining to end, multiple times, various English translations of the Qur'an: Arberry, Pickthall, Rodwell; recently begun studying the hadith; and supplemented the study of the Qur'an with an English translation of Tafsir Ibn Kathir.
The tiny fraction of TED viewers who have any familiarity with Islam will likely have scanned (not read) one of these translations, will never have read a single hadith, and will never have even heard of a tafsir. Further, almost no one realizes the importance of the hadith. The average American would have no idea what a Qur'anist is, and if you told them what it means, they would think all Muslims are Qur'anists.
As for the number, I have carefully combed the Qur'an for the number 72. It is not there. There is not even a hint at there being a limit on the number of virgins you can have.
As for bosomly shaped women, it is important to understand that Christianity still has a strong influence on the way Americans think, even if they aren't Christians. Christianity traditionally has had a rather dim view of sex; note the fact that Catholic priests are required to be celibate, for example. The fact that there is sex in the Muslim Paradise is therefore scandalous. It wouldn't matter much whether your virgins were exceptionally voluptuous. I count seven suras that talk about virgins in Paradise. Presumably Hazleton is referring in particular to Sura 78, which in English is called The Great Event, in verses 31-33:
Arberry says, Lo! for the duteous is achievement / Gardens enclosed and vineyards / And voluptuous women of equal age
Rodwell says, But, for the God-fearing is a blissful abode / Enclosed gardens and vineyards / And damsels with swelling breasts, their peers in age
Pickthall doesn't describe the virgins: Lo! for the duteous is achievement / Gardens enclosed and vineyards / And maidens for companions
The words voluptuous and swelling breasts are the only words that would be heard by most non-Muslim Americans. And it would be taken in a very sensual way. Further, it would make little difference whether this description were included; the key point is the fact that seven surahs indicate that Muslim men will be given an unlimited number of sex partners.
This is a wrong understanding or interpretation of the word "كواعب" (Kawa'ib), see for example in tafsir ibn Kathir:
( كواعب )
أي : نواهد ، يعنون أن ثديهن نواهد لم يتدلين لأنهن أبكار
this word only shows that their breasts are not hanging down, as it would be the case for a woman who had feed much children or an older woman etc..
It also refers or indicates to the fact that they are neither too young nor old etc.. But this is covered by the following word in the verse "atrab أتراب".
English translation here of tafsir ibn Kathir.
In Tafsir al-Kashaf you may find
والكواعب: اللاتي فلكت ثديهن، وهن النواهد
and in at-Tahrir wa tanwir:
والكواعب : جمع كاعب ، وهي الجارية التي بلغت سن خمس عشرة سنة ونحوها . ووصفت بكاعب لأنها تكعب ثديها ، أي : صار كالكعب ، أي : استدار ونتأ
So the emphasize is on the form of the breast (ثدي), and means that a roundness (فلك) can clearly be seen. Like it can be seen at the bone of the heel (الكعب) to which it was compared in the Quran. Ibn Achour added it refers to a girl in the age of fifteen or similar (as this is the age where the breast of a young girl may begin to show up and grow).
See also this fatwa which exactly covers your question.
For part 1:
Of course it has never ben revealed in the Quran, but you may find some ahadith pointing at this amount, like in Jami' at-Tirmidhi, sunan ibn Majah and Musnad Ahmad. Note that this hadith which is qualified as gahrib by Imam at-Tirmdihi, so it is an ahad which has been transmitted in (almost) all levels of the narrator chain by only one person, which IMO makes it a (very) weak narration as stated here, even if as far as I could check the chain of at-Tirmdihi all narrators have a good to very good level of trustworthiness.
Therefore it is widely qualified as hassan by scholars like ibn Hajar and al-Mudhniri and because in some other hadith colletion a similar hadith could be found on the authority of three other sahaba namely Anas ibn Malik in shoab al-Iman of al-Bayhaqi, 'Oqbah ibn 'Amer in at-Tabarani's Musnad a-Shaamiyn and 'Obadah ibn as-Samit in Musnad Ahmad which all quote "72" of the hoor al-'ayn.
This (Arabic) fatwa deals with the number 72 in the hadith, some scholars like Mulla 'Ali al-Qarri said that is the upper bound for the amount of hoor al-'Iyn a martyr could marry, not an indication on a large amount. An other quote now about the part about "the 70 people he could intercede for" according this hadith, is based on tafsir at-tahrir wa-tanwir of 9:80 where Sheikh ibn Achour (see also the hadith to which he refers) said that "seventy times" doesn't mean the exact number but refers to a big amount.
