Yes, see this plot from IEA: https://www.iea.org/data-and-statistics/charts/comparative-l...

This is as of 2018, and one would expect that the grid will continue to get cleaner as time goes on, which means the BEV options will get cleaner as well.

Even in that case, yes, as large plants generate power more efficiently and more cleanly than a bunch of smaller power plants (i.e. engines).

However the major power grids are no longer 100% fossil fuels.

Yes, because ICE efficiency is so bad you'd still probably reduce emissions caused by mover vehicle use by 30-40%.

Yes. If all your cars get the maximum use out of the energy in the petroleum and you provide atmospheric pollution controls on the power plants that eliminate smog/particulates then you have a net win, even if your total energy from petroleum doesn't change.

The main point is that emissions are produced in the power plant, not outside your house.