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This is a pretty broad question, and I'm certainly not an expert in this area, but maybe I can get you pointed in the right direction. It's easy to look under the hood, but maybe (depending on your experience) not so easy to pin down exactly what's being called. Here's how I proceeded:

sage: R.<t> = PolynomialRing(ZZ) # make the kind of ring I want to check
sage: f = R.random_element(); f  # and an element
3*t^2 + t + 65

See what kind of objects they are:

sage: type(R)
<class 'sage.rings.polynomial.polynomial_ring.PolynomialRing_integral_domain'>
sage: type(f)
<type 'sage.rings.polynomial.polynomial_integer_dense_flint.Polynomial_integer_dense_flint'>

You can browse these source files in your local sage install at $SAGE_ROOT/devel/sage-main/sage/rings/polynomial, or you can browse them online (linked from the "Source Code" section of the main Sage Development page).

Of course it helps if you know where to look; multiplication is implemented (often, including this case) by the _mul_ method, so you can check it directly with

sage: f._mul_??

This will display the relevant part of the source code. There, you will see that this particular multiplication is implemented by fmpz_poly_mul -- this is a FLINT method, and you'd have to look at the FLINT source if you want to go further in this hunt . . .

A similar procedure will, I think, get you started with finding the implementations of the other multiplication algorithms you mentioned. Good luck!