Let $(M,\times)$ be a monoid with zero. Let $\Sigma(M,\times)$ be the set of binary operations $+$ on $M$ such that $(M,+,\times)$ is a ring. Let $\sim$ be an equivalence relation on $\Sigma(M,\times)$ defined by $$+_1\sim+_2\iff(M,+_1,\times)\cong(M,+_2,\times).$$

Let's denote the quotient set $\Sigma(M,\times)/\sim$ by $\Sigma'(M,\times)$ and consider the number $$\mathrm{add}(M,\times)=|\Sigma'(M,\times)|.$$

I have several questions about the behavior of this function, none of which I know how to approach.

$(1)$ For an integer $n\geq 0$, is there always a monoid $(M,\times)$ such that $n=\mathrm{add}(M,\times)?$

$(2)$ The same question with the requirement that $M$ be finite.

~~$(3)$ The two previous questions are equivalent if $\mathrm{add}(M,\times)\geq\aleph_0$ for $M$ infinite. Is it true?~~ It is false by Todd Trimble's answer. $(\mathbb Z,+)$ with a zero element adjoined is another example.

$(4)$ Is there an upper bound to the values of $\mathrm{add}(M,\times)$ over all finite monoids with $0?$ What about all monoids with zero?