The earliest text I know of related to Cundi is the Karandavyuha Sutra. In that sutra, a bodhisattva seeks samadhi using the mantra "om manipadme hum." At the end of the sutra, the bodhisattva achieves this samadhi and then seven kotis of buddhas reply in one voice with the Cundi mantra, something like, "namah saptanam samyaksambuddha kotinam tadyatha om cale cule cunde svaha." Probably later, there was the Saptakotibuddhamatr Cundi Dharani Sutra, "Sutra of the Cundi Dharani, the Mother of Seven Kotis of Buddhas." This text is dedicated to the dharani itself and says nothing about any figure named Cundi. The Cundi Dharani is only treated as a dharani by that text (although lauded in the highest terms as is common in dharani sutras).
I think Chinese texts describing Indian iconography in detail do mention Cundi's appearance, with light skin, eighteen arms, and each implement, exactly as in Chinese depictions (the textual descriptions go into far more detail). I say that I think the common Cundi depiction is male because the chest is largely flat, without breasts, and partially exposed. For that matter, most Chinese depictions of Guanyin at temples are also male. Female depictions tend to be "dressed up" (e.g. White Robed Guanyin) because Chinese culture is far more conservative about bare-chested women than Indian.
Strangely, mirrors for the Cundi Dharani are very commonly sold by Chinese online retailers. The use of a mirror is taken from the Cundi Dharani Sutra, which describes kind of a magical practice of reciting the Cundi Dharani in front of a mirror. The Cundi mirrors sold by online retailers tend to be small and decorative, with the mirror surface and the Cundi Dharani in Ranjana script on the front, and a depiction of Cundi and the Cundi Dharani in Chinese on the back.
As for Master Nan's teachings on the Cundi Dharani, I don't think they are close to either Tibetan Buddhism or Shingon, they are just esoteric Mahayana practices as are common in Chinese Buddhism. There is no particular school or institution that they are part of, as esoteric teachings are often not distinguished much from the exoteric. Or if they are, they are just given the generic label, "Esoteric School" (Mizong). You can find some teachings about the Cundi Dharani from Master Nan, and also from one of his western students, William Bodri.
A few links from our own website:http://lapislazulitexts.com/tripitaka/T1077_LL_cundi_dharani
(Cundi Dharani Sutra)http://lapislazulitexts.com/articles/cundi_dharani
(Intro to the Cundi Dharani)