This blog will have things that have already been explored by others but which I need to lay out for myself. Or potentially new ideas which I need to refine (and check if others have not yet theorised on) before posting on the forums.
TLR uses Lerasium for Feruchemy:
Conclusion: The Lord Ruler either uses Lerasium for Feruchemy or some other unclear purpose.
...what happened to that Lerasium bead?! Was it lost forever in the corpse of The Lord Ruler?
Hoid's use of Lerasium:
2. BS suggests Lerasium can grant the power of many shards (though he could also simply say that Hoid has the power of Preservation)
3. It rewrites your spiritual DNA, and there are ways to do really cool things with lerasium that I don't see how anyone would know. Were most Mistborn to just burn it, it would rewrite their genetic code to increase their power as an Allomancer.
5. "Let me first assure you that the element is quite safe. I have found a good home for it. I protect its safety like I protect my own skin, you might say." This suggests that Hoid keeps the Lerasium inside his body, either for Feruchemical purposes or as a spike. However, Hoid is very bad at hurting people, suggesting Feruchemy. Furthermore, this WoB says outright that Hoid does not have Hemalurgy.
Conclusion: Hoid seems to use the Lerasium for Feruchemy, but it's likely something more complicated is going on.
1. Allomancy is drawing Investiture into your (damaged) soul.
2. The metals are a filter. For example, burning Tin means drawing Investiture from Preservation or Ruin (in the case of Atium), and Tin turns the Investiture into sensory enhancement.
3. Compounding is drawing Investiture into your soul and turning it into a nonstandard (non-Allomantic) type of power, by changing a metal's "signature" via feruchemically charging it.
4. Feruchemy is a zero-sum internal magic, Allomancy is a net-gain external magic. Compounding is using the net-gain Allomancy, but turning Investiture into Feruchemical powers rather than standard Allomantic powers.
5. The final step of Compounding is (usually) storing the net-gain, superboosted attribute into another metalmind.
6. Standard Feruchemy stores an attribute given by the user into a metalmind. With this final step, Compounding stores an attribute provided by Preservation/Ruin into a metalmind.
7. The Lord Ruler turned Atium into an Atiummind, storing a small amount of youth in it, then used the Atiummind, drawing Investiture from Ruin's body and turning it into youth rather than foresight. A small bead used right can return someone to childhood. He stored the youth that he didn't have to provide into Atiumminds and then had to constantly draw from them in order not to die. When Vin deprived him of his atiumminds, he aged dramatically (1000 years old) and died.
When Allomancers burn Atium, they are not drawing power from Preservation (or are they...?) but from Ruin's body, i.e. the Atium itself. Does it suggest that Allomantic strength with regard to Atium is independent from how much Preservation is in your sDNA / how strong your connection to Preservation is / how much Lerasium you ate in your life? Does it have more to do with the volume of Atium you're burning? Does a stronger connection to Ruin increase how much power you can get from an Atium bead?
In other words: if you Hemalurgically steal someone's ability to burn Atium, is the ability really weakened? This has to do with whether Atium Allomancy is really end-positive and of Preservation in some way (in which case the ability would be weakened), or whether it's strictly of Ruin (and end-negative somehow), and more power is considered a "loss" (in which case the ability would be... strengthened?).