Does the convex hull C exist for the vectors in V ? so that v1,...,vk are elements of C

Yes, ofcourse it does.

If I write a vector v' as a convex combination of the vectors V, will v' be an element of C as well ?

Most certainly.

The convex hull is the whole triangle.- two random vectors v1 and v2, form a convex set as a line which connects v1 and v2.

Random tetrahedron, which (sides and interior included) forms the convex hull.- three random vectors v1, v2, v3 form a

But note that in more general spaces, the convex hull does not only consist of the set of convex combinations. Remember the Krein-Milman theorem.