Suppose $W$ is a compact bordism between two closed manifolds $M_1$ and $M_2$, so that $\partial W=M_1\sqcup M_2$. Let $g$ be a Riemannian metric on $W$, and let $d$ denote the corresponding Riemmanian distance function. Suppose that $$d(M_1,M_2)=1,$$ where the distance $d(S_1,S_2)$ between two closed subsets $S_1$ and $S_2$ of $W$ is defined to be the infimum of lengths of curves joining a point in $S_1$ to a point in $S_2$.
Consider the function $$d_{M_1}:W\to\mathbb{R}$$ $$x\mapsto d(x,M_1).$$ I would like to know whether a typical value in $[0,1]$ is a regular value of the function $d_{M_1}$. To be more precise:
Question 1: Does the interval $(\frac{1}{2}-\epsilon,\frac{1}{2}+\epsilon)$ contain a regular value of $d_{M_1}$ for every $\epsilon>0$?
Question 2: Suppose $a\in[0,1]$ is a regular value of $d_{M_1}$. Then is the submanifold $d_{M_1}^{-1}\{a\}$ bordant to $M_1$?