The DC collector current is determined by \$R_E\$:
\$I_C = \alpha \dfrac{9.4V}{R_E} \approx \dfrac{9.4V}{R_E}\$
Since you require \$I_C < 1.25mA \$, the constraint equation is:
\$R_E > \dfrac{9.4V}{1.25mA} = 7.52k\Omega\$
The second requirement, maximum output voltage swing, without any other constraint, doesn't fix the collector resistor value.
We have:
\$ V_{o_{max}} = 19.8V - I_C(R_C + R_E)\$
But, the voltage across \$R_E\$ is fixed at 9.4V so:
\$V_{o_{max}} = 10.4V - I_C R_C\$
\$V_{o_{min}} = -I_C * R_C||R_L\$
If you stare at this a bit, you'll see that maximum output voltage swing is 10.4V but this requires that the product \$I_C R_C = 0\$* which is absurd.
Now, if we also require symmetric clipping, then, by inspection:
(1) \$V_{o_{max}} - V_{o_{min}} = 2 I_C (R_C||R_L)\$
(2) \$10.4V = I_C(R_C + R_C||R_L) \$
Looking at (1), note that, for maximum swing, we get more "bang for the buck" by increasing \$I_C \$ rather than \$R_C \$.
Since we have an upper limit on \$I_C\$, (2) becomes:
\$R_C + R_C||R_L = \dfrac{10.4V}{1.25mA} = 8.32k \Omega\$
which can be solved for \$R_C\$.
*unless \$R_L\$ is an open circuit
Your result has no problem, and your procedure is all right. To verify this, you can substitute the \$Z_{b}\$
$$
Z_{b} = \beta r_{e}+[\frac{(\beta+1)+R_{c}/r_{o}}{1+(R_{C}+R_{E})/r_{o}}]R_{E}
$$
into your result and into the textbook's result, and do some simplification, you'll find they give the same result. Cheers!
Best Answer
The "gain" formula as given by you is the transconductance of a transistor in common-emitter configuration WITH feedback.
1.) Without feedback (RE=0) this transconductance is simply $$gm = \frac{hfe}{hie}$$
2.) With feedback we have $$A= \frac{gm}{1+gm*R_E}= \frac{hfe}{hie+R_E}$$
From this, we can derive the loop gain Aloop=-gm*Re. Note that gm=hfe/hie.
3.) Based on the loop gain expression (and knowing that the output resistance will be increased using feedback) we can expect (approximation):
$$Z_{out}= Z_o*(1+gm*R_E)=r_{ce}*(1+gm*R_E)$$
4.) Comment: The above expression for Zout is an approximation because it is derived under the assumption of IDEAL voltage feedback. However, the feedback voltage is provided across RE (not an ideal voltage source). More than that, also the influence of the input circuitry was neglected. However, we should not forget, that during practical operation of this circuit the large value of Zout is paralleled with the collector resitor Rc (which is much smaller that Zout). Hence, the approximation seems to be acceptable.