[EDIT] This replaces my less focused (and more optimistic) attempt at answering both questions. One remark about terminology: it's more standard to refer to the "smoothness" of a conjugacy class than to the "smoothness" of a centralizer. In the algebraic group context, study of a tangent space (Lie algebra) and adjoint action of the group is a natural tool for investigating smoothness.

1) NOTATION: $G$ is a *simply connected* simple algebraic group over an algebraically closed field $K$ of characteristic $p>0$. Its Lie algebra $\mathfrak{g}$ is then the reduction mod $p$ of Chevalley's $\mathbb{Z}$-form of the corresponding simple Lie algebra over $\mathbb{C}$. If $x \in G$, write $C_G(x)$ for its centralizer in $G$ and $\mathfrak{c}_{\mathfrak{g}}(x)$ for the fixed point space of $\text{Ad} x$ on $\mathfrak{g}$. Similarly, if $X \in \mathfrak{g}$, write $C_G(X)$ for the centralizer of $X$ under the adjoint action of $G$ and $\mathfrak{c}_{\mathfrak{g}} (X)$ for the Lie algebra centralizer of $X$. In each situation the two types of centralizer may be called "global" and "infinitesimal".

2) For semisimple elements of $G$ or $\mathfrak{g}$, global and infinitesimal centralizers always agree in dimension (see Borel, *Linear Algebraic Groups*, 9.1). In general, one has natural inclusions $\text{Lie}\, C_G(x) \subset \mathfrak{c}_{\mathfrak{g}}(x)$ and $\text{Lie}\, C_G(X) \subset \mathfrak{c}_{\mathfrak{g}}(X)$
With this set-up, the Lie algebra of $\text{SL}_n(K)$ has a 1-dimensional center $\mathfrak{z}$ (consisting of scalar matrices) precisely when $p|n$; otherwise $p$ is "very good" for type $A_{n-1}$. For other simple types, $p$ is "bad" just when $p=2$ for types $B, C, D$ or $p=2,3$ for types $E_6, E_7, F_4, G_2$ or $p=2,3,5$ for type $E_8$. For good (or very good) $p$, the behavior of all centralizers follows the characteristic 0 pattern. In general, induction on dimension reduces the remaining hard problems to the case when $x=u$ is *unipotent* or $X$ is *nilpotent*: in a simply connected group like $G$, the centralizer of a semisimple element is reductive and also connected.

3) The center $\mathfrak{z}$ is well understood (Hurley) along with the full ideal structure and its behavior under the adjoint action of $G$ (Hogeweij, Hiss). Beyond type $A$ one always has $\dim \mathfrak{z} \leq 1$ (with equality sometimes for bad primes) except for type $D_{2n}$ where $\dim \mathfrak{z} =2$ if $p=2$.

4) The answer to the first question is almost certainly no: all known methods for the study of the various centralizers when $p$ is not very good for $G$ involve case-by-case study of the simple types. What is known so far makes it unlikely that more uniform methods will emerge.

5) In type $A_{n-1}$, comparison with the well-behaved centralizers in $\text{GL}_n(K)$ shows that centralizer dimensions go up by 1 in $\text{Lie}\, \text{SL}_n(K)$ for unipotent or nilpotent elements just when $\mathfrak{z} \neq 0$ (i.e., $p|n$).

6) In other types, the literature seems inconclusive about further unusual contributions to $\mathfrak{c}_{\mathfrak{g}} (u)$. (The new AMS monograph
here by Martin Liebeck and Gary Seitz promises to deal systematically but still case-by-case with many fine points about classes and centralizers for unipotent elements in the group and nilpotent elements in the Lie algebra.) So far the emphasis has been on classification and counting of unipotent classes or nilpotent orbits for bad primes, along with explicit descriptions of centralizers in $G$. In fact, it's unclear to me what consequences a resolution of the problem posed by Steinberg in his (12) might have (rephrased here as the second question). Steinberg himself looked directly at the case of a *regular* unipotent element in section 4 of his 1965 IHES paper on regular elements (see 4.3), where global and infinitesimal centralizers differ at most by $\mathfrak{z}$; but this requires delicate *ad hoc* arguments.

7) In his 1966 ICM survey, Steinberg's format is stimulating and unusual: he gives a concise list of 28 items, including summaries of recent work along with 18 items labelled "Problem". These are of varying breadth or precision but are by now mostly resolved one way or the other. Problem (10), later resolved by Lusztig, involves the finiteness of the number of unipotent classes in all simple algebraic groups; this isn't as "modest" as Steinberg expected it to be. On the other hand, problem (22) was over-optimistic. No doubt it would be useful for someone to write a short follow-up survey on the status of all those problems. (In my own 1995 AMS book *Conjugacy Classes in Semisimple Algebraic Groups* I aimed mostly for the uniform results in good characteristic. The book lists most of the relevant earlier references.)