Yes, the traces pretty much follow the grid, though if you use parts with different lead spacing (such as SMD parts, with lead spacings measured in mm, used on a grid that is a fraction of inches) it will let a section of the track be off the grid, usually it snaps back to the grid as soon as you put a bend in the trace, etc.
Both the regular and alt grids are adjustable, apparently the alt grid is just there if you need to use a smaller grid occasionally, like for squeezing traces between pads, etc. I normally use a grid of .0125 inch.
You can set the grid in inches, mils, mm, or microns. the trace widths, hole sizes, etc, are in the same units as you set your grid.