1)"Wide Supply Range:VS= ±4.5 to ±24 V" You only have +5V and 0V.
Choose a low voltage part not a high voltage part.
2) "Common-mode input range" Supplys-3 volts. In this case; 5-3=2 and 0=+3=3 so there is no input voltage that will make the parts work because the supply is too low. You need a input range that includes ground and maybe close to +5V.
So look for a RR input (rail to rail).
3) This part will output the 10mA you want but it will take most of the 5V supply to do that. (it will not deliver 10mA with an output voltage of 4.5V as suggested in post #5) You will need to have a 0 ohm resistor to get this output with this opamp.
4) This part does not need to be so fast.
Sorry for the bad news.