Why isn't this water softener shutoff valve configuration preventing water from running in my house
In my experience, quarter-turn ball vales are extremely reliable. I've seen some incredibly rusted out plumbing, including where the valve handle has completely fallen off after rusting through, and you can still reliably shut the valve with a pair of pliers or replacement handle.

It's hard to check if it's faulty 100% (without cutting pipes, anyway), but as a first step:


Disconnect the softener. 


It looks like you have the right head to be able to do this. Turn the valves off, then turn the handle on the front to drain (may want to do it a couple times until you hear no more water draining) -- if you skip this you will get quite wet, trust me. Unscrew the two metal clips on each side, then the softener should pull straight forward -- there are o-rings providing the seal. 

Alternatively, turn the dial to Drain and listen to the water flowing in the next steps. This might be a bit less reliable but should still help.

Open the supply side valve a bit, you should get water coming out.
Open the service side valve a bit, you should not get any water out (other than maybe what drains out).


If you do, SOMETHING is wrong: the bypass valve is faulty OR there is a cross-connection. By operating the bypass valve, you might be able to tell if the pressure increases (indicating cross-connection) or does nothing (faulty valve) but this isn't 100%.  
If nothing comes out, then with the service side open and supply side closed, open the bypass a bit just to verify that water comes out. 

With service side open, and supply and bypass valves closed (there should be no water coming out of the softener connections), check the rest of the taps in the house. Report back here, I can't even really speculate what's happening in this case, other than perhaps your softener is not actually connected at all.