Your first question is hard to answer as posed. In general, one can say very little about what kind of singularities a particular fiber might have even if the total space and hence the nearby fibers are non-singular.

For example, let $X_0$ be the projectivized cone over a hypersurface of degree $d$ in $\mathbb P^n$. Then $X_0$ is itself a degree $d$ hypersurface in $\mathbb P^{n+1}$, and hence a fiber in a family with smooth general fiber. However, $X_0$ has log canonical singularities if and only if $d<n$, so in most cases it does not.

In general, one wouldn't expect a condition that guarantees that a fiber has slc singularities, but a procedure, often called *stable reduction*, that can be applied to most families to obtain such a family. Well, OK, with stable reduction you get a little more, but this can be tweaked.

Anyway, the way you *make* the fibers have slc sings is to apply the relative mmp to your family and that produces nice singularities. Depending on your goal you might stop at the relative minimal model or go on to the relative canonical model. For the latter you would (currently) need additional assumptions, but if you are trying to do/use moduli theory then you're OK.

For your second question, perhaps the best to do is to use Du Bois singularities. The following is true:

- slc singularities are Du Bois
- Gorenstein Du Bois singularities are slc.

In other words, for Gorenstein singularities slc is the same as DB.

From the way you posed your question it seems that you assume that $\Delta$ is a disk in $\mathbb C$. In any case, if $\Delta$ is smooth, then every fiber in $X$ is a complete intersection. So, one possible answer to your second question is that if $X$ has Gorenstein singularities (actually a little less is enough) then what you are asking is true. This follows from Theorem A of KS16 and the fact that a general hyperplane section of a DB sing is DB cf. Prop 6.2 in Kollár13