Scott, it looks like you're trying to edit the STL mesh, having imported it back into Rhino? You can do that but you'll be much better off editing the original Rhino surface model and then converting it to a mesh and exporting the mesh as an STL. You edit surface models using the points as well, Although, unlike when working with meshes, there are many more powerful and versatile editing and modeling tools available for surfaces.
A quick note - Using Grid Snap is virtually useless for most things, unless you need to confine points to a particular grid spacing. The Gumball is a pretty powerful tool as you can enter exact amounts of movement.
An alternative to Julian's way to move points a specific distance is to select the points (or and object you want to move), type MOVE, enter 0 as the "Point To Move From" prompt and ENTER, then at the Point to Move To prompt type in the XYZ coordinates of the move, for example entering .5mm, 0, 0 and ENTER would move the selection .5mm along the X axis. If the UNITS are already set to millimeters you don't have to type the "mm", just the number. Keep in mind that all of my files were modeled in inches so you'll have to convert them if you want to work metric.
Rhino is primarily a Surface Modeling program. Because so many people are working with mesh scan data, they did make it possible to edit meshes with a fairly limited suite of tools. And while there are Mesh Boolean functions (Boolean commands are the backbone of and good 3D modeling program) Rhino's Mesh Boolean functions do not work reliably, if at all. On the other hand, Rhino's surface model boolean commands rock! Very powerful, very flexible.
You asked about stretching a face or plane... you can stretch a part of an object but you can't stretch a single face, al least not perpendicular to the face (what's know as it's Normal). I also use the Scale1 command a lot to stretch object along a single axis.
Another way to make such an adjustment is to Extract the surface, move it, loft a new surface between to fill the gap (or build manually if a loft isn't possible or practical) and then join all back together into a Closed Polysurface. This will only work with surfaces, not meshes.
Something you will learn with Rhino is that there are usually several different methods of achieving a desired result, all of which will create identical objects, just getting there along different paths. Some are more efficient than others, some more logical, some less prone to error but more steps, and some just become your favorite way of working.