Try tapping the Z button twice. You should see the axis line change. One tap* on the Z key moves along local axis, the second tap* on the Z key glides along the global ("true") axis.
Think of it like in Astronomy with time and sidereal time. It's the exact same thing.
TL;DNR: You seem to have done some of your rotations in object mode and some in edit mode. That's why setting rotation is "working". It has nothing to do with the keyframes, except that trying to create them is exposing the problem. The "fix" in the case of a plane is to rotate in normal space.
You are not being stupid, you've just come ...
EDIT: so I've just realized that all your cones are supposed to converge towards the same point, in that case, as Robin says, you can just press E to extrude then press Enter right away, and scale down with S0. When all your cone tips are on the same point, merge by distance with AltM
A simple Extrude will extrude along the Z local axis of the selected faces....
A metal or wooden beam is to be stretched up in an arc with it's ends firmly clamped.
(Blend file below)
A lot of problems can be avoided when using the Laplacian modifier by ensuring the mesh object(s) it's going to be used on and those associated with it, are all scaled to 1. (CTL-A)
Similarly, hooks used to deform it should also have a scale of 1 ...
The Y axis is set by the (normalized) vector from bone head to bone tail. To change this axis, move the bone tail in edit mode. It may be necessary to remove the "connected" trait from children bones to do this, which is fine; connected trait is nearly useless.
Once you have the Y axis set, X and Z axes can be set by bone roll.
Whenever you ...
This can be done via the UI.. it's relatively destructive, but here goes..
Enable the shipped add-ons Add Mesh : Extra Objects, and Bool
Set your Pivot to 'Individual Origins', and your Transform
Orientation to 'Normal'
Add a Mesh > Math Function > Regular Solid > Dodecahedron
ShiftG Select Similar, by Length, unwanted edges, and CtrlX ...
TBH not totally sure I'm on track here, re symmetry of desired result, will go thru the process of adding as dupliface objects, aligned to the face normals
Enable Add Mesh Extras Addon and add the dodecahedron from the presets Add Mesh > Math Functions > Regular Solids
Add a dodecaheron, dissolve the long edges crossing the pentagon