Another, more intuitive, take on your question (mathematicians: please look away) is to think of the dot (A.B = ABcosθ) and cross (AxB = ABsinθ) products as simply ways to measure degrees of parallelism of vectors versus perpendicularity (orthogonality) of vectors, in the sense that:
The dot product of parallel unit vectors, U.U yields a number U.U cos 0 = 1 (or a number of value A*B for vectors of arbitrary vector lengths) while of orthogonal (perpendicular) vectors it’s always zero, as cos 90°=0
Conversely, the cross product of parallel unit vectors is of magnitude 0 (as sin 0°=0) while of orthogonal unit vectors it’s 1 (or magnitude AxB for arbitrary lengths); but in this case the result is mapped into a vector, which needs to be perpendicular to the plane defined by the input vectors A and B (there’s no obvious way to assign it a direction within the plane defined by A and B). And recognise also that the cross product thereby gains a sense of handedness as AxB = -BxA, which turns out to be useful (example below).
Of course, it’s the use of sin and cos that determines the way these measures range from 0 to 1 for unit vectors; just imagine the products' values changing as you think of the A and B vectors rotating towards or away from each other for each type of product.
As far as physical significance is concerned (and ignoring the deeper insights available via Clifford Algebra etc.), these “products” turn out to be useful in so many situations that their physical significance is often taken for granted, rather than being emphasised (and maybe this underlies your question).
For the dot product: e.g. in mechanics, the scalar value of Power is the dot product of the Force and Velocity vectors (as above, if the vectors are parallel, the force is contributing fully to the power; if perpendicular to the direction of motion, the force is not contributing to the power, and it's the cos function that varies as the length of the projection of the force vector on the velocity vector varies; so it's not at all arbitrarily defined).
For the cross product: e.g. angular momentum, L = r x p (all vectors), so it seems perfectly intuitive for the vector resulting from the cross product to align with the axis of rotation involved, perpendicular to the plane defined by the radius and momentum vectors (which in this example will themselves usually be perpendicular to each other so the magnitude of rp*sin90°= rp). And if the direction of rotation changes, the sign of the momentum vector is reversed and so the cross product vector L also changes sign (hence the usefulness of mapping the cross product into a vector).
Note however, you can also calculate the number resulting from AB*sinθ (rather than mapping it into a perpendicular vector). It’s just the area of the parallelogram that is defined by the vectors A and B in the cross product.
Incidentally, there's nothing stopping you mapping the dot product into a perpendicular vector, if so desired - but it's probably not often useful to do so in physics.
As regards division, that’s a little more technical and dealt with well by the earlier replies. There’s also some approachable discussion on https://www.quora.com/Can-we-divide-a-vector-by-a-vector-and-why
I hope this is of some help for the less-expert members.