I don't have a reference for all of these, but the answers can be worked out from the APS theorem. Note that you should be a little careful with the notation; these are really metric invariants, and one usually works with metrics that are products near the boundary.

For the first, to discuss $\eta(M \times N)$ then $M \times N$ is supposed to be an odd-dimensional closed manifold of dimension $4k-1$, so let's suppose $M$ has odd dimension and $N$ has even dimension. I claim that $\eta(M \times N) = \eta(M) \sigma(N)$ if dim(M) is of the form $4j-1$. The answer is $0$ if dim(M) is of the form $4j+1$.

Let's assume also that $M$ is an oriented boundary, say $M = \partial W$. (In general, you know that some number, say $n$ of copies of $M$ is a boundary, and you apply the argument those $n$ copies.) Then $M \times N= \partial(W \times N)$ and you can use the multiplicativity of the L-class and the signature (this is true for manifolds with boundary) plus (or should I say times!) the signature theorem applied to $N$ to get the result. The other case is easier, since the signature of $N$ is $0$ and the L-class of the product is 0 for simple dimensional reasons.

For the second, the answer is definitely no. Indeed, much of the second APS paper is devoted to the $\rho$ invariant, which roughly speaking measures the difference between $\eta(\tilde{M})$ and (degree of the covering times)$\eta(M)$. The $\rho$ invariant is in an appropriate sense a topological invariant; perhaps this is what you are referring to in reference to distinguishing homeomorphisms types.

For the third, $\eta(2M) = 0$ since $2M$ (with a metric obtained by gluing together identical metrics on the two copies of $M$) has an orientation reversing isometry. This point is discussed in the introduction to the first APS paper.