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This is basically a long comment.

I will say a little bit about restrict my remarks to your first question, except that I will talk about controlling cut/focal points instead of intuitively understanding them, but I hope some intuition comes from the remarks. Maybe you already know the standard facts that will follow.

I think that cut points are much more mysterious and harder to understand than focal points. In the presence of curvature bounds, focal points can be controlled pretty well due to comparison geometry. Let us talk about focal points to embedded submanifolds. For the comparison space, you take a manifold of constant curvature K, and then construct an embedded submanifold by requiring that all eigenvalues of the second fundamental form be (the same) constant $\lambda$. You can write down the distance to the focal locus of the submanifold explicitly in this case in terms of $K$ and $\lambda$. Now given an arbitrary riemannian manifold with curvature bounded above by $K$ and second fundamental form below by $\lambda$, you can favorably compare the distance to the focal locus to that of the comparison space, see Warner, F. "Extension of the Rauch Comparison Theorem to Submanifolds" http://www.jstor.org/pss/1994552

In general there is no lower bound for the cut locus distance of a submanifold even if there are curvature bounds. But there are results if the submanifold is convex in some sense. For instance, consider a manifold with boundary $(M, \partial M)$ with mean curvature $H > 0$ and suppose that |Ric$(M)| \leq K$. Then we can use a second variation argument (that only applies if we haven't yet reached the focal locus) to put a lower bound on the cut locus distance in terms of the dimension of the manifold, $K$, and $H$. This lower bound goes to infinity as $K \to 0$, so if your curvature bound is small enough you can try to say things like the first cut point cannot occur before the first focal point.

Finally, I will just point out that it is easy to make examples of manifolds with boundary that have arbitrarily small cut locus distance but no focal points. Consider $\mathbb{R}^3$ and let $B_1$ be the solid ball of radius $1 - \epsilon$ centered at $(0,0,0)$ and let $B_2$ be the solid ball of radius $1 - \epsilon$ centered at $(2,0,0)$. Let $(M, \partial M) = \mathbf{R}^3 - B_1 - B_2$. Then the boundary has no focal points but the cut locus distance is $2\epsilon$.

Hope it helps a little bit. Sorry if

EDIT: I am just pointing out obvious stuffsee my comments may be too general, given that your question was specifically focused on curves and surfaces. For a surface $S \subset \mathbf R^3$, the ambient manifold has curvature zero so there is a nice relationship between the curvature of $S$ and the second fundamental form. My general feeling is that the cut points should be related to issues of convexity, while focal points are more related to "how much $S$ is curved."
show/hide this revision's text 1

This is basically a long comment.

I will say a little bit about your first question, except that I will talk about controlling cut/focal points instead of understanding them. Maybe you already know the standard facts that will follow.

I think that cut points are much more mysterious and harder to understand than focal points. In the presence of curvature bounds, focal points can be controlled pretty well due to comparison geometry. Let us talk about focal points to embedded submanifolds. For the comparison space, you take a manifold of constant curvature K, and then construct an embedded submanifold by requiring that all eigenvalues of the second fundamental form be (the same) constant $\lambda$. You can write down the distance to the focal locus of the submanifold explicitly in this case in terms of $K$ and $\lambda$. Now given an arbitrary riemannian manifold with curvature bounded above by $K$ and second fundamental form below by $\lambda$, you can favorably compare the distance to the focal locus to that of the comparison space, see Warner, F. "Extension of the Rauch Comparison Theorem to Submanifolds" http://www.jstor.org/pss/1994552

In general there is no lower bound for the cut locus distance of a submanifold even if there are curvature bounds. But there are results if the submanifold is convex in some sense. For instance, consider a manifold with boundary $(M, \partial M)$ with mean curvature $H > 0$ and suppose that |Ric$(M)| \leq K$. Then we can use a second variation argument (that only applies if we haven't yet reached the focal locus) to put a lower bound on the cut locus distance in terms of the dimension of the manifold, $K$, and $H$. This lower bound goes to infinity as $K \to 0$, so if your curvature bound is small enough you can try to say things like the first cut point cannot occur before the first focal point.

Finally, I will just point out that it is easy to make examples of manifolds with boundary that have arbitrarily small cut locus distance but no focal points. Consider $\mathbb{R}^3$ and let $B_1$ be the solid ball of radius $1 - \epsilon$ centered at $(0,0,0)$ and let $B_2$ be the solid ball of radius $1 - \epsilon$ centered at $(2,0,0)$. Let $(M, \partial M) = \mathbf{R}^3 - B_1 - B_2$. Then the boundary has no focal points but the cut locus distance is $2\epsilon$.

Hope it helps a little bit. Sorry if I am just pointing out obvious stuff.