Why are currents IQ2 and IQ3 in this circuit the same, but is IQ1 larger (image)
Because \$I_{Q1}\$ consist of the collector current of Q1 plus the base currents of Q1+Q2+Q3.


  All bases of the bipolar transistors are connected to the same current. I would expect all three currents to be the same.


This is not correct. All bases are having the same voltage.
If the transistors are identical, the you can indeed except the same collector currents for Q1, Q2 and Q3 because they share the same \$V_{be}\$