We know that a Type II$_\infty$ factor can be written as a tensor product of a Type II$_1$ factor and a Type I$_\infty$ factor. Is there any explicit example of II$_\infty$ factor constructed this way?
Second question: what is the algebra of the tensor product of a Type II$_1$ factor and a Type I$_n$ factor?
Not entirely sure what you mean by "explicit" here. Take any II$_1$ factor $N$, any infinite-dimensional Hilbert space $H$, and form $M=N\otimes B(H)$. Then $M$ is II$_\infty$.
As for your second question, for a II$_1$-factor $N$ you have a canonical isomorphism $$ N\otimes M_n(\mathbb C)\simeq M_n(N). $$ And it's easy to see that $M_n(N)$ is a II$_1$-factor for any $n\in\mathbb N$.