Fewest number of gates for Multiplication What is the best result for the number of gates in a circuit multiplying two n-bit integers?

The obvious method generates $\theta(n^2)$ gates. There are better approaches with $\theta(n\log n \log\log n)$ and $\theta(n\log n2^{\log^*(n)})$ gates.

I could not find any Boolean circuit family which can handle multiplication with $n\log n$ gates. I wonder if such a family of circuits exists.
Below is a detailed 2008 survey that covers top theoretical algorithms for multiplication, including the ones discussed in the comments to your question (including the Schönhage–Strassen algorithm and the $O(n\log n \, 2^{\log^* n})$ Fuerer algorithm, see page 335 of survey). However, implementation is a different matter and some of these algorithms may not be considered practical; the survey does not cover practical implementations. Although the survey includes algorithms for polynomials, power series, real numbers, and 2-adic numbers, integers are a special case of these (see Figure 1 on page 336).

Fast Multiplication And Its Applications, Bernstein (Algorithmic Number Theory / MSRI Publications / Volume 44, 2008)