What happens when you give a Leadership character with Song of Wisdom a Burning Brand and then choose to discard Song of Wisdom from a card effect? I believe I have read somewhere the Brand still stays but that seems to go against the passive text rule.

More confusingly, what happens when you place Squire's Helm on an ally but then Sword-thain the ally? Is Squire's helm discarded? I assume it has no more effect regardless because it says " Attached ally gets +2 hit points."

The same can be asked about Valiant Determination.

RR Attachments :

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Play restrictions, such as "attach to a Gondor hero" are checked only at the time the attachment enters play.

As you state, many attachments will lose their effects if the character loses the appropriate type or trait, though.

(I am ever amused that Dark Knowledge loses its negative effect on willpower but retains its Response if the attached hero becomes an ally, e.g., Spirit Imrahil.)