Tiiba: "(Smoke implies fire, therefore fire) implies (no smoke means fire)?"
"therefore fire" should be left out.
i)
(Smoke implies fire) implies fire.
ii)
No smoke implies fire.

You're confused because i) is false.

Medium puzzle:

"Applying the Deduction Theorem to Löb's Theorem gives us, for all C:

((◻C)->C)->C"

Wouldn't it be:
((?C) -> C) -> ?C
If the provability of C implies C's truth, then C is provable.

In other words, where you write "(X->Y)->Y", it should be "(X -> Y) -> X".

Tiiba: "(Smoke implies fire, therefore fire) implies (no smoke means fire)?" "therefore fire" should be left out. i) (Smoke implies fire) implies fire. ii) No smoke implies fire.

You're confused because i) is false.

Medium puzzle:

"Applying the Deduction Theorem to Löb's Theorem gives us, for all C:

Wouldn't it be: ((?C) -> C) -> ?C If the provability of C implies C's truth, then C is provable.

In other words, where you write "(X->Y)->Y", it should be "(X -> Y) -> X".