Matched biasing transistor
I'll answer the two parts of the questions (and thank Andy Aka for stimulating me to make my answer clearer).

Answer to first part of the question: it is not ok to short the \$10\mathrm{k\Omega}\$ base resistors.  The circuit behavior changes significantly, in the sense that if you replace the  with shorts, the resistive part of the input dynamic impedance of the circuit will drop down. Precisely, if you do it the circuit become the following one: 



 

simulate this circuit &ndash; Schematic created using CircuitLab 

Now the dynamic impedance seen immediately after the capacitor \$C\$ is equal to 


the input resistance of \$Q_2\$ in parallel with
a \$20\mathrm{k\Omega}\$ resistor and with 
the series of the impedance of BJT \$Q_1\$ with collector shorted to base and a \$180\Omega\$ resistor. 


The dynamic impedance of BJT \$Q_1\$ so connected is 
$$
r_{Q_1}=\frac{V_T}{I_B+I_C}\approx\frac{25\mathrm{mV}}{0.96\mathrm{mA}}=26\Omega,
$$
being 
$$
I_B+I_C\cong\frac{20\mathrm{V}-V_{BEQ_1}}{20\mathrm{k\Omega}+180\Omega}\cong\frac{20V-0.7\mathrm{V}}{20\mathrm{k\Omega}+180\Omega}\cong 0.96 \mathrm{mA}
$$
Therefore the full input dynamic resistance seen from the input capacitor is less than \$r_{Q_1}+180\Omega\approx 206\Omega\$, due to the parallel connection.

The aim of the resistors in series to the base of each matched transistor is exactly to keep the input impedance of the stage of the order of several \$\mathrm{k}\Omega\$: the \$10\mathrm{k\Omega}\$ in series to the base of \$Q_1\$ may seem superfluous in this context, but it keeps as close as possible the bias voltages and currents of the two BJTs, exploiting fully their matched characteristic.

Answer to the second part of the question: the stability of the circuit itself is not affected by the shorting of the base resistors, since there is no intentional feedback from the input of the circuit to its output (if we assume that \$C_{cbQ_1}\$ is negligible at the normal operating frequencies). However, in this case the circuit loads in a stronger way any previous stage, thus it can cause system stability issues, if the preceding stage is not carefully designed.