What does direction of travel have to do with time dilation?
v is the
relative velocity in the Lorentz relation. As you see, the answer is the same whether
v is positive or negative.
No, space is space, of units "meters" (cubic meters) and time is time (of units "seconds"). Both are relative, and inversely so, which makes your claim all the more impossible.
No, space and time (in relativity) are related by the Lorentz transformation. It's a coordinate transformation (projection) of a rotation (in this case hyperbolic). This has little or nothing to do with the Pythagorean theorem, and your reference to complex numbers seems to deny the relationship between hyperbolic and standard trig functions, which you are not going to disprove no matter how hard you try.
which is nothing more than the product of 4 individual standard rotations, substituting the hyperbolic functions for the standard trig functions.
Time and space don't "behave". Nor did anyone say that lightspeed is completely defined by anything. That being said, there is reason to conclude that lightspeed (geometric mean of the electric permittivity and magnetic permeability of the vacuum) can be attributed to the statistical lifetimes and and densities of Fermion pair creation/annihilation:
http://www.physics.utah.edu/~lebohec/Speed_of_light/QuantumVaccumAsOriginOfC2013.pdf
and as you see the permittivity and permeability are similarly "created".