> Could we use this technology to completely eradicate from the world all species of mosquito that prey on humans?
Yes, implemented correctly, a gene drive has this capability.
> Also, could we accurately predict the extent of the resulting ecological disruption so we could decide if it was worth it?
The current scientific and political consensus is that, no, we can't predict the potential consequences with high enough confidence to move forward with such a large-scale manuever.
One of the leading scientists studying and improving gene drives, Kevin Esvelt at the MIT Media Lab, instead supports a "daisy chain gene drive" which is pre-programmed to weaken with each successive generation, allowing humans to effectively control the spread of gene drives to a specific geographic area and time frame.