Do we in general have $$\min_x \left[f(x) + \min_{y} g(x,y) \right]= \min_{x,y} f(x) + g(x,y),$$
i.e. we choose some $x$ which gives us an $f(x)$, and then, treating $x$ as constant, we minimize $g(x,y)$ in $y$. Is this equivalent to choosing $x, y$ simultaneously to minimize $f(x) + g(x,y)$?
Seems obvious, but maybe it isn't?
Yes, since $$\min_{x,y} \left(f(x) + g(x,y)\right) = \min_x \left(\min_y(f(x) + g(x,y))\right) = \min_x (f(x) + \min_y g(x,y)) \, .$$