Convexity of log det X???

Basically Boyd is taking the approach of showing what happens with $\log \det$ in a neighborhood of $t=0$. You're quite correct in that you can make $Z+tV$ into something that's not positive definite, but that isn't really the point here. In fact if you let $t=0.1$, then the sum is still p.d.

What this approach is doing is to show that if you have a line through any $Z$, then you can show that it's concave along that line. That doesn't mean that you have to follow that line out of the domain under scrutiny. It just gives you a way to show convexity locally.

As for $V$ being a "direction": this might be a little confusing until you realize that matrices can be vectors, and that $S^n$ is just a vector space. $Z+tV$ is just the expression of the fact that $Z$ is a vector, and by adding $tV$ (also a vector), you're moving in a line through $Z$. $V$ can be any symmetric matrix, because the important thing is that $Z$ is a member of the domain you're looking at ($S^n_{++}$, the domain of $\log \det$).

So assuming $t$ is small enough, you can answer the question of whether $g$ is concave on a line through $Z$. The fact that $S^n_{++}$ is an open subset of $S^n$ lets us do that.