Necessity of beta in common emitter design
Why is β ever used in this kind of analysis?


In this particular bit of analysis, for normal small-signal transistors (i.e., \$\beta \simeq 100\$ or so), the value of \$\beta\$ really only enters into things in the choice of values of R1 and R2.  Assuming zero base current, all you need to do is get the ratio of R1 and R2 correct.  But a real base current will pull \$V_{be}\$ down.  So you want to, first, take that into account, and second, don't make R1 and R2 too large.

My rule of thumb is that the quiescent current in R1 should be about ten times the expected base current at room temperature (and, thus, that the expected standing current in R2 is 90% of that in R1).  My other rule of thumb is that if I'm doing this for real and not just some on-bench experimentation, I check the circuit operation over temperature.


  ...you should design them so the voltage across RE is 1 V. Why?


First, that's a rule of thumb, so it's yours to violate.  Second, that gives a pretty good all-around stabilization of the emitter current over part to part and temperature variations.  To be sure, again, you want to check your circuit (usually by Monte Carlo simulation) over variations in part parameters and temperature.