Let the zero sets $F=\{x \in \mathbb{R}^n: f(x) = 0\}$, $G = \{x \in \mathbb{R}^n : g(x) = 0\}$, where $f$ and $g$ are $m$-dimensional real, analytic, continuous, and nonlinear vector functions. Under some assumptions, these sets define hypersurfaces of zero Lebesgue measure in $\mathbb{R}^n$. I was wondering:

1. Are $F$ and $G$ always submanifolds embedded in $\mathbb{R}^n$ or are there exceptions - in the latter case, are there conditions that guarantee that they are submanifolds ?
2. What is the dimension of $F\cap G$ ? As pointed out here Measure of the intersection of two manifolds, if $F$ and $G$ are $(n-1)$-dimensional manifolds and their intersection is transversal, then $\text{dim}(F\cap G) = n-2$. However, is there anything that can be said if the intersection is not transversal ? In general, I am interested in some sort of inequality $\text{dim}(F\cap G) \leq n-2$, assuming that $F \subseteq G$, that $G \subseteq F$ do not hold.