Is it permissible for my fiancée to give me a small lock of her cut hair as a keepsake, and does the ruling differ if she dyed its color, and what is the legal impediment to that?

What falls from a person's hair is desirable to bury. As for the ruling on looking at a woman's detached hair, while scholars differ on the ruling of looking at it or touching it, preserving and honoring it is not required. It is safer to avoid looking at detached hair from a non-mahram woman, and a fiancée is considered a non-mahram to her fiancé until the legal marriage contract is concluded.