What does it say under that scratched drawn tree in the third panel?

That's an umbrella. Typically, you write your name and your lover's name under the canopy, much like the American practice of putting your initials inside a heart (usually carved into a tree).

Since I don't read Japanese hardly at all, all I can guess is it's their names. The implication being, since Aya's in love with Momiji here, she wrote the umbrella thing... And Momiji scratched it out.

Those of you who CAN read Japanese, feel free to correct me if I'm wrong.

Honestly can't make out the first symbol, but the rest reads みじ for mizi or miji. The other one clearly reads Aya-san.