Let $S$ be a commutative $\mathbb{C}$-algebra and let $a,b,c,d \in S$. Assume that $R_1:=\mathbb{C}[a+c,b+d] \subseteq S$ is flat and $R_1:=\mathbb{C}[a-c,b-d] \subseteq S$ is flat.
Question 1: Is it true that $\mathbb{C}[a,b,c,d] \subseteq S$ is flat?
More generally, if $A_1,A_2 \subseteq B$ are two subrings of $B$ (sub $\mathbb{C}$-algebras of the $\mathbb{C}$-algebra $B$) such that $A_i \subseteq B$ is flat, $1 \leq i \leq 2$,
Question 2: Is it true that $B$ is flat over $A$, where $A$ is the $\mathbb{C}$-algebra generated by $A_1,A_2$?
Non-counterexample: $A_1=\mathbb{C}[x], A_2=\mathbb{C}[y], B=\mathbb{C}[x,y,z]$; all the relevant extensions are free, hence flat.
Counterexample: $A_1=\mathbb{C}[x^2], A_2=\mathbb{C}[x^3], B=\mathbb{C}[x]$: Here $A_i \subseteq B$ are free (hence flat), but $A=\mathbb{C}[x^2,x^3] \subseteq B$ is not flat. (If we change the first question to $R_1=\mathbb{C}[a], R_2=\mathbb{C}[b]$, then the ring generated by $R_1$ and $R_2$ is $\mathbb{C}[a,b]$ and $\mathbb{C}[a,b] \subseteq S$ is not flat for $a=x^2, b=x^3, S=\mathbb{C}[x]$).
Question 3: Is it possible to find a mild condition which will guarantee positive answers to my questions?
The condition: $R_i \subseteq S$ are integral ring extensions will not help, as the above counterexample shows.
What about a condition involving fields of fractions?
What about a condition saying: $A_i \subseteq B$ is a separable (or even etale) ring extension?
This is a slightly more general question, with no answers. This question is relevant (especially, if we further assume that $S$ and $B$ are integral domains). Tensor products and products of rings (probably less relevant).
Now also asked in MO.
Thank you very much!