Calculating base current without knowing current amplification
That`s quite normal.
You are choosing a suitable DC operational point (collector current Ic, resistors Rc and Re).
The voltage gain depends on the transconductance gm (slope of the voltage-control function Ic=f(Vbe) at the DC operating point). The so-called &quot;current gain&quot; has no direct influence on the voltage gain.
For calculating the resistive base divider values you must make a rough estimation for the DC current gain B (in case you do not have any value).
As a first step you may assume an infinite B value (Ie=Ic).
In many cases, this is sufficient if the divider current is much larger than the (unknown) base curent. Very often, the error caused by such a simplification is even smaller than the error caused by parts tolerances and by the rough estimation Vbe=0.7 V.
And yes - it is correct that the input resistance at the base is part of the overall input resistance.
However, if you have no information about the actual base current you have no other choice than to make a good estimate.