An opamp circuit schematic

a): An answer:
B
b): An answer:
$\frac{i_{x}}{v_{in}} = \frac{g_{m}}{2}$
c): An answer:
$r_{in} = 2 \frac{βfe}{g_{m}}$
d): An answer:
$i_{x} = i_{y}$
e): An answer:
$r_{in} = R_{L} (1 + βfe) + \frac{βfe}{g_{m}}$
f): An answer:

$\begin{array}{l} v_{out} & = R_{L} \cdot i_{e} \\ i_{e} = g_{m} \cdot (v_{b 3} - v_{out}) \cdot \frac{βfe + 1}{βfe} \\ v_{out} & = R_{L} \cdot g_{m} \cdot (v_{b 3} - v_{out}) \cdot \frac{βfe + 1}{βfe} \\ = v_{b 3} \cdot R_{L} \cdot g_{m} \cdot \frac{βfe + 1}{βfe} \cdot \frac{1}{1 + R_{L} \cdot g_{m} \cdot \frac{βfe + 1}{βfe}} \\ \frac{v_{out}}{v_{b 3}} & = \frac{R_{L} \cdot g_{m} \cdot (βfe + 1)}{βfe + R_{L} \cdot g_{m} \cdot (βfe + 1)} \end{array}$
g): An answer:
$r_{out} \to \infty$ .
h): An answer:

$\begin{array}{l} a_{v} & = \frac{g_{m}}{2} \cdot 1 \cdot (R_{L} (1 + βfe) + \frac{βfe}{g_{m}}) \cdot \frac{R_{L} \cdot g_{m} \cdot (βfe + 1)}{βfe + R_{L} \cdot g_{m} \cdot (βfe + 1)} \\ = \frac{g_{m} R_{L} (1 + βfe)}{2} \\ \approx \frac{g_{m} R_{L} βfe}{2} = 10000 \end{array}$