shivaniits,
Your post is long, but I'll take a stab at addressing at least part of it. I have no idea what "flux lines" are, other than another (very poor) name for "field lines." The fields of classical electromagnetism are vector fields: mathematical functions that assign a vector to every point in space. As a result, they can be depicted by drawing arrows at each point in space (with a chosen sampling interval) to indicate the magnitude and direction of the field at that point. An alternate way to depict them, however, is to join up the arrows to form smooth, continuous curves called "field lines" that indicate the overall structure of the field. Information about the strength is not lost, because for a consistent choice of field line density (i.e. how many of them you draw for a charge of a given strength) the strength of the field is determined by how close together or far apart the field lines are. Imagine field lines radiating outward from a point charge: the field is stronger closer to the charge where the lines are denser and weaker farther away where they are more sparse. However, the flux through a closed surface around that charge, which is represented by the total number of field lines passing through it, is independent of the distance to that surface. This makes sense, because although the field strength diminishes as 1/r^2, the area of the surface increases as r^2, and the flux, which depends on the product of these two, is therefore constant. If you have a chance, you should check out section 2.2.1 in
Introduction to Electrodynamics by David J. Griffiths, where he explains this better (pp. 6567 in the third edition).
As far as "flux density" goes, its important to keep in mind that these are names, and the choice of names is sometimes arbitrary or historical. Just because the name has the word "flux" in it doesn't mean it is supposed to be the same type of quantity as the "flux" that you know of, that is defined in terms of a surface integral. Words are just words, and this wouldn't be the first instance of the same word being used in two different ways in physics (not by a long shot). After all, "flux" is just Latin for "flow." In any case, the two quantities are not of the same type. The "flux" of the electric field and the "flux" of the magnetic field, (##\Phi_E## and ##\Phi_B##) are scalars, whereas the quantity that some people refer to as the "magnetic flux density"
B is unquestionably a vector. As I stated before, in terms of mathematical definition, the fields of electromagnetism (
E,
B,
D,
H, take your pick) are all
vector fields. EDIT: I do see now from you original post that some sources justify the use of the term "flux density" for
B, by noting that when you integrate it over an area, you get the magnetic flux. Fair enough.
For what it's worth, Griffiths balks at the choice of "magnetic flux density" as a name for
B
Quote by David J. Griffiths in Introduction to Electrodynamics
Many authors call H, not B, the "magnetic field." Then they have to invent a new word for B: the "flux density," or "magnetic induction" (an absurd choice, since that term already has at least two other meanings in electrodynamics). Anyway, B is indisputably the fundamental quantity, so I shall continue to call it the "magnetic field," as everyone does in the spoken language. H has no sensible name: just call it "H."^{4}

^{4}For those who disagree, I quote A. Sommerfield's Electrodynamics (New York: Academic Press, 1952), p.45: "The unhappy term 'magnetic field' for H should be avoided as far as possible. It seems to us that this term has led into error none less than Maxwell himself..."
