Why my non-inverted amplifier does not work?
The datasheet will tell you what the common mode input voltage can be. From memory, you can't go closer than 4V above the negative supply and I think you need at least 10 V between the two supplies.
You can use a wide range of resistors. Very low values may overload the op-amp output and very high values may lead to excessive phase shift at high frequencies, offsets due to input bias current, and high noise.