In 2D, they are not so weird as you think. Write the vectors as complex numbers, $2+3i$ and $5+3i$ (where $i$ is the usual imaginary number, not your $\vec i$). The "ordinary" product is
very close to the product of the first number by the conjugate of the second,
where you recognize both the dot- and the cross-products !
By the way, you say that for addition we are just "adding magnitudes", but this is not true, the magnitude of the sum can be smaller than the sum of the magnitudes.
This whole "complication" (which is in fact a wonderful source of richness) comes form the fact that vectors need not have the same direction and do not add/multiply like scalars.