Op amp controlled LED current
The chosen opamp MCP6241 is quite suitable for this application as it is rail-to-rail input/ouput, and Vcc as low as 1.8v. You have two options to handle it: 1) add a resistor between Vcc and inverting-pin of opamp. Because when the + pin is zero, the - pin tries to reach zero as well but because of input-offset, an error appears on the output. so you have to make sure that with zero input on + pin, there's a &quot;near-zero but above-zero&quot; voltage on  - pin.  Choose it's value as high as possible, for example in range of 10Kohm, a value so much bigger than the the resistor in series with LED. Also, you's better add a capacitor between - pin and GND, in case of any potential instability. 2) add a high value resistor in parallel with the LED.  I suppose the latter solution works better. To solve your second problem you could use a lower value for the resisor R; and also add a voltage divider (resistor-divider) in front of + pin. In this case a fraction of input voltage is applied to the + pin. So your input voltage range can be 0 to Vcc, while, for instance, 1/5 of this range appears on + pin and consequently on - pin, which provides a good range of current on LED.