Classification of "complete dense ordered near-semirings"

Question

Let's define a complete dense ordered near-semiring, or a CDON, as a set $A$ equipped with two binary operations $+,\times$ and a binary relation $\leq$ such that:

1. $+$ is a monoid, whose identity is denoted by $0$;
2. $\times$ is a monoid, whose identity is denoted by $1$;
3. $(a+b)c=ac+bc$ and $c(a+b)=ca+cb$ for all $a,b,c\in A$;
4. $0a=0$ and $a0=0$ for all $a\in A$;
5. $0\neq1$;
6. $\leq$ is a dense total order which has the least upper bound property;
7. For all $a,b,c\in A$, if $a\leq b$, then $a+c\leq b+c$ and $c+a\leq c+b$;
8. For all $a,b,c\in A$, if $a\leq b$, then $ac\leq bc$ and $ca\leq ca$.

Question 1. Is $+$ cancellative? That is, if $a+c=b+c$ or $c+a=c+b$, then does it follow that $a=b$?

Question 2. Does all nonzero $a\in A$ has a multiplicative inverse?

I think the above two questions are true because of the least upper bound property. If $A$ is indeed cancellative, then $+$ becomes commutative since $a+a+b+b=(a+b)(1+1)=a+b+a+b$, where $a$ and $b$ can be cancelled. But I do not know about $\times$:

Question 3. Assuming that $A$ has the properties mentioned in questions 1 and 2, is $\times$ commutative?

Question 4. Can we (partially or fully) classify CDONs?

If $A$ has all the properties in questions 1-3, then $\mathbb Q\cap[0,\infty)$ can be embedded in $A$ in a unique way. And I think it will follow that $A$ is isomorphic to $\mathbb R\cap[0,\infty)$. I feel that questions 1 and 2 is true, and 3 is false, so that there is one commutative CDON and some other noncommuative CDONs.