Let's start with the second set of questions, because that partially explains what happens in the first.

**Why [is] the singularity at the vertex of $C_V$ the same as  $\mathbb C^3/(x∼−x)$?**

Well, the Veronese is the image of the map taking the degree $2$ monomials in $3$ variables, so its homogenous coordinate ring is the ring $\mathbb C[x^2,y^2,z^2,xy,xz,yz]$, which is exactly the invariant subring under the action $(x,y,z)\sim (-x,-y,-z)$, so the cone $C_V$, which is just $\mathrm{Spec}\ \mathbb C[x^2,y^2,z^2,xy,xz,yz]$, will be the quotient of the affine space by that action.

**And [why does] the modification the exceptional locus ha[ve] pure dimension $2$?**

This is true for all $\mathbb Q$-factorial singularities. See [this answer][1] to another MO question. 

So let's get to the first set of questions: 

**How [can one] describe the blowing up $C_F$ along $C_l$[?]**

The funny thing is that what makes this work is that the singularity of $C_F$ is *worse* than that of $C_V$ in the sense that it is not $\mathbb Q$-factorial (this is because the Picard group of $F$ has rank $2$ as opposed to the Pcard group of $V$ being $1$). Because of that it admits a *small morphism* (see the linked answer for details). The non-$\mathbb Q$-factoriality means that there is a Weil divisor which is not $\mathbb Q$-factorial. In this case that's $C_l$, so blowing it up you get a small morphism. 

**[W]hy does the special fibre $C_4$, after blowing up, have only Du Val singularities?** and **What is the strict transform of $C_4$?**

The above allows for the fibers to remain normal and shows that the strict transform=preimage of $C_4$ is just the blow up of $C_4$ at its vertex which is obviously Du~Val and nothing else changes, because $C_l$ is Cartier everywhere else.



  [1]: https://mathoverflow.net/a/45193/10076