Why do these two single supply noninverting amplifers not work but the third does?
Circuit 1 doesn't work mostly because your 100k pair voltage divider is not appropriate to set a bias point for the inverting (-) feedback network.
The currents flowing through the 4k &amp; 1k will easily pull that so-called voltage divider waaay off where you thought it would be and this is what gives you a gain of approximately 1.1
Also, your input to the non-inverting (+) input of your opamp has no DC bias set - it depends entirely on the opamp's biasing/leakage currents.   

Circuit 2 doesn't work because you're applying a DC bias to your non-inverting input (good), but then amplifying it by the gain of your opamp circuit (bad). So your opamp is trying to drive its output to 25V, but it only has a 10V supply.  

Circuit 3 works because you've separated the DC gain from the AC gain by adding that capacitor.
This circuit has an AC gain of 5, but a DC gain of only 1, so your input signal appears on the output amplified by 5 and riding on a 5V DC level.