I power-leveled a character to 81, and gave her a full complement of Stealth perks. Additionally, she has some perks added to Smithing and Enchantment.
Then, I began taking her on a crime spree. Pickpocketing, breaking and entering, burglary... whatever I could do to get her some stolen gold. Here's what I've gotten so far.
Stolen Gold Test
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Container Amount Marked? Notes
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Chest 32 No Whiterun - Ysolda's House
Pickpocket 9 No Whiterun - Olava the Feeble
Coin Purse 20 No Whiterun - Olava the Feeble's House (Cupboard)
Chest 17 No Whiterun - Olava the Feeble's House
Cupboard 14 No Whiterun - Uthgerd's House
Wardrobe 5 No Whiterun - Uthgerd's House
Chest 32 No Whiterun - Uthgerd's House
Cupboard 1 No Whiterun - Carlotta Valentina's House
Chest 32 No Whiterun - Carlotta Valentina's House
Chest 35 No Whiterun - Heimskr's House
Pickpocket 6 No Whiterun - Danica Pure-Spring
Pickpocket 46 No Whiterun - Nazeem
Wardrobe 8 No Whiterun - House Grey-Mane
Cupboard 7 No Whiterun - House Grey-Mane
Chest 11 No Whiterun - House Grey-Mane
Chest 9 No Whiterun - House Grey-Mane
End Table 5 No Whiterun - House Grey-Mane
Pickpocket 6 No Whiterun - Avulstein Grey-Mane
Pickpocket 13 No Whiterun - Amren
Pickpocket 11 No Whiterun - Ahlam
Cupboard 9 No Whiterun - Jorrvaskr
Coin Purse 28 No Whiterun - Jorrvaskr Living Quarters
Cupboard 8 No Whiterun - Jorrvaskr Living Quarters
Wardrobe 12 No Whiterun - Jorrvaskr Living Quarters
Wardrobe 6 No Whiterun - Jorrvaskr Living Quarters
Coin Purse 60 No Whiterun - Jorrvaskr Living Quarters (WITNESSED)
Coin Purse 56 No Whiterun - Jorrvaskr Living Quarters
End Table 6 No Whiterun - Jorrvaskr Living Quarters
Coin Purse 50 No Whiterun - Jorrvaskr Living Quarters
Chest 11 No Whiterun - Jorrvaskr Living Quarters
Coin Purse 22 No Whiterun - Jorrvaskr Living Quarters
Chest 21 No Whiterun - Jorrvaskr Living Quarters
Chest 21 No Whiterun - Jorrvaskr Living Quarters
Coin Purse 54 No Whiterun - Jorrvaskr Living Quarters
Coin Purse 32 No Whiterun - Jorrvaskr Living Quarters
End Table 5 No Whiterun - Jorrvaskr Living Quarters
Coin Purse 44 No Whiterun - Jorrvaskr Living Quarters
Where the Container is a Coin Purse, those are just coin purses lying about loose somewhere. For Containers marked Pickpocket, these are coins taken directly from an NPC via pickpocketing. See the Notes column for further details on all.
The Marked column denotes whether or not the gold ended up marked as stolen in my inventory. All of the gold documented above was marked in red, or flagged with the "Steal" action when I acquired it.
I've also done a bit of testing on some other suggestions.
- Pickpocket, die, reload, kill & loot does not seem to work.
- Reverse-pickpocket & re-pickpocket doesn't seem to do the trick.
Best Answer
I have done a lot of testing on this and have yet to find a way for a companion to actually learn a spell. If you give them staves that have spells on them they will spam them. As long as they are not conjuration spells. I used Lydia and when given conjuration staves she will revert to her bow and will not even equip the stave that summons. I am also unable to unequip her bow and arrows. They seem to be "Basic" companion items they always have.
So in short: No they cannot learn spells.
Although: If your companion is a spell caster they will cast the spells they already had but will not learn more.