Definition of simplex

A simplex (in $\mathbb{R}^{n}$) is a special case of a convex polytope, namely one with exactly $n+1$ extremal points (the convex hull of $n+1$ points in general position - that is to say they don't lie in a $(n-1)$-dimensional affine subspace). An example of a convex polytope which isn't a simplex is given by any one of the platonic solids except the tetrahedron.

Yes, there are more general notions of simplices. In a geodesic metric space, for example, it makes sense to speak of convex hulls. In hyperbolic $n$-space or on the $n$-dimensional sphere one can then define a hyperbolic or spherical simplex as the convex hull $(n+1)$ points that don't lie in a totally geodesic submanifold (and that are not too far apart from each other in the case of a sphere).

The definition is a little unclear. In order for the convex hull of $n+1$ vertices in $\mathbb{R}^n$ to actually be $n$-dimensional, the vertices must be in general position: that is, they should not lie in a proper affine subspace of $\mathbb{R}^n$. For example, a square is a convex polytope which is the convex hull of $4$ vertices, but it is not a $3$-simplex: a $3$-simplex is a tetrahedron.

I suppose the answer to the last question is "no": when people talk about geometric simplices they are definitely referring to the ones in $\mathbb{R}^n$, although some people talk about abstract simplices in the sense of abstract simplicial complexes.

I often work over simplices in $\mathbb{F}_q^d$ or other discrete spaces. Here, when we say a $k$-simplex, we mean a set of $(k+1)$-points spanning a $k$-dimensional subspace. I.e., for us, a $k$-simplex is just the set of vertices.

No, they are not equivalent. For example, an $n$-cube is a convex polytope but is not a simplex.