What is the most natural way to express these ideas in French, along the lines of the do-support in English or ellipsis sentences?
- Do you want to go to the store?
Yeah, I do!
Don’t forget to take out the trash.
- I already did.
In French, to say, "yes, I do," you'd simply say "oui." If you want to emphasize that you really do want to go, you could use "bien" as a modifier for intensity (but you'd have to include the original phrase from my understanding):
As far as your second example, a neuter pronoun exists:
The l' attached to "ai" is the neuter pronoun. Without it, the literal translation is "I already did," but whom you're speaking to could very well respond with, "tu as fait quoi ?" meaning, "you did what?" So the neuter pronoun is necessary to avoid any imprecision.
I hope this helps!