Why MOSFETs in Battery Management System are connected to negative terminal?
Because the N-channel MOSFET can carry higher current and it is cheaper then its similar size counterpart P-channel MOSFET. N-ch MOSFET can be used also as high side switch, but it requires an additional charge pump.

The shunt is placed on lower side, because the common mode voltage would be too high for the operational amplifier. Again, it is possible to measure the current on high side, but you would require an isolated DC/DC converter and isolation operational amplifier or some specific opamp designed for high side sensing. You can find high side sensing and switching in expensive industrial equipment. 

The price is the main reason.

INA138 - high side current sense