There are a number of very simple operations under the Edit tab, that essentially have no parameters. You won't see a property panel for these tools, they immediately apply their result to the scene.
The Duplicate operation makes copies of all the selected objects. The copies will be given the same name with (copy), (copy 1), (copy 2), ... as suffixes. This is the exact same operation as the Duplicate button in the Object Browser. The hotkey for the Duplicate command is shift-c.
There is not a specific menu item in the Edit tab for the Discard operation, which deletes all the selected scene objects (Meshes, Pivots, etc). However you can use the delete key or the x hotkey to Discard.
Combine and Separate Shells
The Combine operation merges all the selected Meshes into a single Mesh object. However, Combine is not a Boolean!! The triangles of the meshes are simply appended, so overlapping objects will still be overlapping regions of triangles. New Face Groups are assigned for each input object.
The Separate Shells operation finds all disconnected regions of triangles of the active mesh - we sometimes call these connected components - and creates a new mesh object for each region. In some tools these are referred to as "shells" and we have copied that terminology here. This operation discards the original mesh, so if you need to keep it around, do a Duplicate first.
The hotkey for Combine is y, and for Separate Shells it is shift-y.
The example below shows a single mesh (left), which appears from this viewpoint to be a single shell, but is actually two overlapping shells (middle-left). Separate Shells will turn this single self-intersecting mesh into two separate Meshes (middle-right), and Combine will merge them back together (far-right).
So, in this case Combine and Separate Shells are "opposite" commands, however this is not always the case! If you Combine two Meshes that each have two shells, and then do Separate Shells, you will get four Meshes, not two!
The Close Cracks tool solves a few irritating problems that arise in mesh-based modeling. One has to do with imported meshes that contain disconnected internal boundaries. In these cases you will see thin blue "crack" edges, where the crack actually has identical boundary loops on either side, but they are topologically separated (for example many OBJ files with texture seams are created this way). The Close Cracks tool will weld these cracks together, if possible. However it does not do any complex stitching, and if welding vertices would involve any non-trivial movement or create non-manifold geometry, the crack will not be closed.
Close Cracks is also incredibly useful if you would like to apply operations to a sub-region of a mesh. Meshmixer does not have masking functionality, and some tools can only be applied to entire mesh objects. So, if you would like to work on a sub-region, you can select that region, use Separate to isolate it, apply the operation, then Combine with the original object and Close Cracks. The only critical requirement is that the operations you apply not change the mesh boundary (otherwise you will no longer have a clean "crack").
This is also good way to work on the "inside" of meshes, that you otherwise would not be able to see (some day we will have clipping planes....). It will also let you isolate sub-regions of meshes that are otherwise too large to work with due to slow rendering updates.
The images below show how to use this workflow to isolate a region for 3D sculpting. The ear is Separated, sculpted, and Combined, and then Close Cracks merges the meshes into a single continuous shell. Note that during the sculpting the Fixed Boundary filter must be checked to precisely preserve the boundary loop.