All the pickups for acoustics (which include the T5) are in-house Taylor designs, AFAIK. From 2003, the incarnation of the Expression System (aka the "ES" amplification on the acoustics) were in-house designs. The famous Rupert Neve was purportedly brought in for guidance on the preamp system. The under-fingerboard pickup on the ES acoustics resembles a thin humbucker that is affixed beneath the fretboard extension, though since the T5 employs an exposed neck pickup I think it
doesn't have one of these. Yet the body sensors
are borrowed from the ES' parts bin. I am not privy as to whether they are the same exact units, but in principle they are the same body sensors that transduce the soundboard's "tone" to the preamp.
I've not ever seen any pics of inside the T5, but that would be cool. I wonder, too, if the factory has ever published any as I've never seen a single gut-shot of a T5, but they have often revealed gut shots of acoustics over the decades.
So in sum and to your questions:
1. Pickups are not from other brands as they are wholly Taylor designs
2. Not mics, which is why the T5 is so feedback resistant.
3. Not piezo crystal as Taylor designed their own transducers for the body tone. First piezo use was the "ES2," and even then it is a unique Taylor design and not the ubiquitous UST piezo strip.
Hope that helps you a bit

Edward