If a creature's toughness is reduced to 0 or less by Yahenni's Expertise, that creature is still on the battlefield as you cast your free spell. It can be the target of that spell, although it will become an illegal target before that spell resolves. If it has any abilities that trigger on the spell being cast, those abilities will trigger.

A card's converted mana cost is determined solely by the mana symbols printed in its upper right corner. The converted mana cost is the total amount of mana in that cost, regardless of color. For example, a card with mana cost has converted mana cost 3. Ignore any alternative costs, additional costs, cost increases, or cost reductions that could apply to it. A card with no mana cost has a converted mana cost of 0.

If the card has in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.

Effects that allow you to "cast" a card don't allow you to play a land card.

If you cast a card "without paying its mana cost," you can't pay any alternative costs, such as emerge costs. You can, however, pay additional costs. If the card has any mandatory additional costs, such as that of Cathartic Reunion, you must pay those to cast the card.

While you're casting your free spell, the Expertise spell is still on the stack. It will be put into its owner's graveyard after the free spell is cast. The free spell can't target the Expertise card in your graveyard. It can target the Expertise spell on the stack, but the Expertise spell will become an illegal target before the free spell resolves.

Any triggered abilities that trigger while performing the Expertise spell's first effect won't be put onto the stack until after you're done casting your free spell. They're put onto the stack at the same time as any abilities that triggered while casting that spell regardless of the order in which those abilities triggered.