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I can't speak for the FFT claim, but least the sub-O(n log n) multiplication claim is for the multitape Turing machine model with a fixed number of tapes, which if anything is more restrictive than actual random-access memory. Of course, it's still galactic just like the O(n log n) algorithm, but it definitely seems like there's something there.

You can be linear in the bit length of the input number, but exponential in the magnitude.

This is a common thing in complexity, and I think even the abstract says it is talking about “length n” and not “magnitude n”

No, because in the model of computation they're using, you can multiply any two numbers in constant time. See https://enwp.org/Real_RAM for a similar model.