I am trying to understand some points about the paradox, what I am doing is solving the for the step potential $$V = V_0 ~\theta(z)$$

I have two solutions $$\Phi_I = e^{ik_1 z} + r e^{-ik_1 z}$$

for $z<0$ $$\Phi_{II} = t e^{ik_2 z}$$ for $z>0$,

where

$$k_1 = \sqrt{E^2-m^2}$$ $$k_2 = \pm \sqrt{(E-V_0)^2-m^2}$$

The first question is, why there is no $\pm$ from the square root for $k_1$, why is not possible for $k_1$ to be negative

Why is possible for $k_2$ to be negative for strong potentials (that is $V_0 > E^2+m^2$) making $$T=\frac{4k_1 k_2}{k_1+k_2}<0$$ therefore showing the paradox with this nonsense but it won't happen for weak or intermediate potentials (Thats $V_0 < E^2-m^2$ and $E-m < V_0 < E+m$) even when the $\pm$ it's always there.

They may be silly questions, but I just don't see it.

For the first question, the answer is: in this context we just want to study the case where an electron wave is coming from $-\infty$ to the barrier, so we just consider a positive $k_1$. One can of course also consider an electron coming from $\infty$ to the barrier, then one just uses $k_1=-\sqrt{E^2-m^2}$.