The ability that defines Tymaret's toughness works in all zones, not just the battlefield.

Because damage remains marked on a creature until the damage is removed as the turn ends, nonlethal damage dealt to Tymaret may become lethal if your devotion to black decreases during that turn.

Colorless and generic mana symbols ({C}, {0}, {1}, {2}, {X}, and so on) in mana costs of permanents you control don't count toward your devotion to any color.

Mana symbols in the text boxes of permanents you control don't count toward your devotion to any color.

Hybrid mana symbols, monocolored hybrid mana symbols, and Phyrexian mana symbols do count toward your devotion to their color(s).

If an activated ability or triggered ability has an effect that depends on your devotion to a color, you count the number of mana symbols of that color among the mana costs of permanents you control as the ability resolves. The permanent with that ability will be counted if it's still on the battlefield at that time.

If you put an Aura on an opponent's permanent, you still control the Aura, and mana symbols in its mana cost count towards your devotion.