Simultaneous power supply from battery and USB
1) When I connect USB, then the 5V from the USB goes to SMPS output.
  Is that ok? It has EN pin, but I don't know.


The datasheet for the AOZ1280 says "Do not leave the EN pin floating." In the case where the 6-12V supply is disconnected, that may cause an issue if the USB is powered - you could take a large value resistor from VIN to GND to ensure that EN is not floating, at the expense of increased battery drain and complexity.


  2) What happens if a user connects both USB and battery at the same
  time? Especially what happens If there's computer plugged to the USB
  and I connect battery?


Any difference in the voltage outputs will result in large currents flowing between the supplies. In general, connecting two supplies together is bad, even if they are nominally the same voltage.


  3) Do you know any ICs that could multiplex power supply? Something
  like: I connect USB PS and battery PS to the chip, it detects which
  one is present (or has control inputs) and send one of them to the
  output.


There are many methods to do this, both with ICs and discrete components. This answer seems to summarise it nicely.


  I would like to avoid using diodes because of voltage drop (first
  power loss, second 5V won't be 5V any more).


You can increase the output voltage of the AOZ1280 by changing the feedback resistors. This would at least eliminate the "5V won't be 5V" problem.