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How can I make the base of the pyramid coplanar to (smooshed up against, so they are flat against each other) the slanted face of the wedge (pictured below)?

enter image description here

  1. The objects and their origins need to stay the same.

  2. If all of the pictured geometry were part of one object, and #1 was not a requirement, how could this be achieved? (If this were a mesh level transformation instead of an object level transformation)

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Multiple objects:

For this you can use custom transform orientations again:

  1. Select the face and press ⎈ Ctrl⎇ AltSpace:

    enter image description here

  2. In object mode, select the object you wish to align and press Align to transform orientation in 3D view > Header > Object > Transform:

    enter image description here

  3. Once the object is aligned, use transform snapping with face as the snapping element.

    If the bottom of the spike thing is already the closest face to the target, then closest should be fine as the snapping target. If not, use the 3D cursor trick as explained here.

    enter image description here

I think you should be able to use transform snapping for rotation too, but for no apparent reason there is an inexplicable gap when snapping without a pre-aligned model..

enter image description here

I've reported this as a bug.

Single mesh:

A similar approach to this can be used:

enter image description here

A custom transform orientation aligned to the target face will help too.

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