Not sure if this was mentioned already, but have you tried turning off the transform gizmo when moving/snapping your vertices? The hotkey for this used to be X. With the gizmo out of the way, the start snap point is on the vertex itself, not where ever you happen to click on the gizmo. And you can change the axes faster…
You can make your own "grid object" in 3ds max and activate it so that your grid-snap points will snap to it rather than the usual orthographic or ground-grid. Just create a grid object somewhere. Right click to activate it, and And you can set the size and planes and color and so on. Right click again and click "activate…