Hausdorff, spherical Hausdorff and dyadic net measures not only give rise to the same dimension but, for a fixed value of $m$, are comparable up to constants that depend only on the ambient dimension $d$. In particular, the property of having zero, positive and finite, or infinite measure coincides for these three measures.
I am sure that there are examples (even fairly simple examples such as self-similar sets) for which the actual values of these three measures differ, but I don't have a concrete example or reference at hand.
As you say Hausdorff dimension is the most used, and the reason to consider spherical or dyadic net measures is that they sometimes make calculations simpler (while yielding the same notion of dimension, and even of zero/infinite measure).
One very useful way to study the size of sets is by representing them as trees, and a universal way to do so in Euclidean space is via dyadic cubes - when doing this dyadic net measures are more natural. For example, the standard proof of Frostman's lemma (one the most basic results about Hausdorff measures) uses dyadic partitions and dyadic net measures, although it is often stated for Hausdorff measure.
On the other hand, covering by spheres are often easier to analyze than covering by arbitrary sets, and for this reason spherical measures is sometimes easier to compute (if one is seeking the exact value). In particular, the study of densities (the behavior of $\mu(A\cap B(x,r))/r^m$ as $r\to 0$, where $\mu$ is some measure of interest) is easier for spherical measures
For nice regular sets (such as embedded manifolds) all three measures do agree, so I don't think there are examples where size according to spherical or net measures is "totally wrong".
I hadn't heard of Gross or Caratheodory measures before, but here are some general remarks. I'll denote Gross measure by $G_m$, Hausdorff measure by $H_s$ and Lebesgue measure by $L_m$.
Since they are defined only for integer $m$, it does not make too much sense to consider a dimension associated to Gross measure, but in some sense it agrees with the notion of Hausdorff dimension. If $\dim_H A>m$ (where $A\subset\mathbb{R}^n$ and $\dim_H$ is Hausdorff dimension), then $G_m(A)>0$. This follows from the Marstrand-Mattila projection theorem, that says that if $\dim_H A>m$, then the projection of $A$ onto almost every $m$-dimensional subspace has positive $L_m$-measure. In fact it is enough to know this for only one projection. On the other hand, $G_m$ is bounded above by a constant multiple of $H_m$ (since projecting does not increase Hausdorff measure, and on an $m$-dimensional subspace, $H_m$ is a constant multiple of Lebesgue measure). In particular, if $\dim_H(A)<m$, then $H_m(A)=0$ and so $G_m(A)=0$.
However, it is possible that $H_m(A)>0$ but $G_m(A)=0$. Indeed, there exist sets of positive and finite $m$-dimensional measure such that their projection onto any $m$-dimensional subspace has zero $L_m$-measure. For example, this is the case if $A$ is a self-similar set satisfying a suitable separation condition if the orthogonal parts of the generating similitudes generate a dense subgroup of the orthogonal group (this was proved by Eroglu in the plane and Farkas in arbitrary dimension, but ad hoc examples were known long before). For my intuition, it is "correct" to say that such self-similar sets have positive $m$-measure, so Gross measure looks "wrong" to me here - but this is likely simply because I'm much more familiar with Hausdorff measure.
Everything I've said about Gross measure applies equally well to Carathéodory measure.
Best Answer
Yes. The boundary even has a locally finite Haudsorff $(m-1)$-measure.
No. A convex function of 1 variable has increasing derivative, but this derivative can have a dense set of jumps.
In general, the function describing the boundary is only Lipschitz (and differentiable almost everywhere).
For all these facts, you may consult a nice book Hormander, Notions of convexity, Chap II.
On your other questions. Of course, there is no reason to consider Hausdorff measure of a convex set: it is ordinary Lebesgue measure in the linear span of this set. I guess the paper you mention considers Hausdorff measure on the BOUNDARY of a convex set. As I said in 1, it has integer dimension. But so what? It is not a smooth surface. What other measure you propose to consider on it?