1. Yes, this is how CADs worked for ages.
2. Cursor does snap to the grid, the component center does not. Let's say I have a 0.1" grid and a resistor that is currently at a coordinate (1.15, 3.21). If I select it, I can move it to coordinates (1.25, 3.21) or (7.15, 3.61). But I can't move it to (1.1, 3.2).
So, mouse snaps to the grid. But if you are moving a component that is already not on the grid, it never will be on the grid with this approach.
Once I manually edit component location to be on the grid, all following movements are working as expected.