How are hybrid parameters defined?
For any linear two-port, we can define the h-parameters according to the equation
$$\begin{pmatrix}v_1 \\ i_2\end{pmatrix} =
  \begin{pmatrix} h_{11} &amp; h_{12} \\ h_{21} &amp; h_{22} \end{pmatrix}
  \begin{pmatrix} i_1 \\ v_2 \end{pmatrix}$$
In the case of the BJT in the common-emitter configuration, we're taking \$v_1\$ to be between the base and common (emitter) terminals, and \$v_2\$ to be between the collector and common terminals. According to the passive current convention, the port currents \$i_1\$ and \$i_2\$ are then the currents in to the base and in to the collector, respectively.

Shouldn't Δv_BE = Δv_be as V_BE is constant (DC) and Δv_be is variable (AC), how did it become just v_be (same for the other currents/voltages).

This is just a matter of notation chosen by the author of your textbook. It's common to use upper case \$V\$ for the DC voltage and lower case \$v\$ for the ac small-signal variation about the bias point. But I'm not aware of any convention that says the subscript must also be lower case for the small signal parameter.

In each paramter why did the condition variable (ex: v_CE = constant in h11) go from being a constant to being 0.

Because \$h_11\$ (for example) is telling you about the response of \$v_1\$ due to stimulus from \$i_1\$ (\$i_B\$). If you want to know how \$v_1\$ varies in response to stimulus at \$v_2\$ (\$v_{CE}\$) you need to look at the \$h_{12}\$ parameter instead of \$h_{11}\$.

Shouldn't the variables shown on the circuits be the total ones not just the AC variables (v_eb should be v_BE)?

No, the h-parameters can only describe a linear circuit, or, in this case, a linear approximation to the circuit's behavior when driven with a small-signal stimulus about a bias point.
The response of the &quot;total&quot; variables like \$V_{BE}\$ and \$I_C\$ isn't linear, so it can't be described by h-parameters.