Hi again!
Paul, I used a 0.1 ohm resistor because it was easier to come by and as ivan747 pointed out, power loss isn't very big. I also estimated that the voltage drop when charging (50mV @ 0.5Amp) is negligible as I currently use a slow charger. Thanks for your tip about LM358, as I don't need four channels.
NANDBlog, thank you for a very thorough look and explanation, I promise to be more doubtful of the simulation software (Multisim). You exposed a problem of two different grounds. I tried with the grounds joined and the results were pretty much alike, huge negative voltage on U2B. Then, yes, I've been avoiding instrumentation amplifiers because I've never dealt with them before, but I'll give a look at them now and the datasheet you recommended.
Thanks to all, I'm learning a lot!