The above mentioned long exact sequence also exist in cases where $V$ and $V'$ are nice enough topological spaces. However in that case it is only an exact sequence of vector spaces. The main reason to restrict to singular algebraic varieties is that even in this case one might construct examples such that $b_2(V')-b_2(V)>1$ holds.

A. Grassi, D.R. Morrison, Group representations and the Euler characteristic of elliptically fibered Calabi-Yau threefolds. J. Algebraic Geom. 12 (2003), 321–356.

... Depending on your base type of equation etc. you should play around a little bit with the equations in order to find a birational model for your situation, where your threefold is a complete intersection of ample hypersurfaces (this should not be too hard, since you assumed that your ci is relatively ample) and where you can control the singularities (this is more annoying).

If you want this particular example, go to arxiv look for a paper written by Klaus Hulek and me (there are only two such papers) and download the first version of the correct paper (rather than the most recent version). The method is a little ad hoc. I.e., in our example we started with $Y^2Z=X^3+aXZ+bZ^3$, where $a,b$ were in some $H^0(S,L^4)$ and $H^0(S,L^6)$ resp. and $X,Y,Z$ are some choice of vertical coordinates. Now you can consider $a,b$ also as elements from $K(S)=K(P^2)=k(s,t)$, homogenizing $a$ and $b$ yields a hypersurface $y^2=x^3+ax+b$ in some wps...

Yes! Blowing up this point gives the image of the zero section.

In Griffiths-Harris you find a description for the cohomology of a projective bundle. In particular, if h^i(B) is nonzero then h^i(P(E)) is also nonzero.