Analyzing BJT differential pair with feedback
That's not a capital "B", but a beta (&beta;).  It refers to the gain of the transistor, collector current divided by base current.

You can consider the 1 mA evenly distributed as a first order approximation.  In reality, this only happens right at the cusp where the differential pair is in balance.  A little off from that point, and one transistor will rapidly take more of the current than the other.

However, this circuit puts negative feedback around this differential pair, always keeping it close to the balance point.  With the input at ground, the other input of the diff pair will also be held close to ground.  That requires close to 10 V across the 20 k&Omega; resistor, which means the right transistor of the pair is drawing &frac12; mA, so the left one is drawing the other &frac12; mA.