How can you go from DINER to MENUS, changing one letter at a time and forming another English word in each step?

Which of the following statements are true and which are false?

1. Only one of the statements is false.
2. Exactly two of the statements are false.
3. Only three of the statements are false.
4. Exactly four of the statements are false.
5. All five of these statements are false.

Rearrange the letters of the first word to form a new one, then rearrange the letters of the new word to form a third.

ASLEEP _ _ _ _ _ _ _ _ _ _ _ _
PARTS _ _ _ _ _ _ _ _ _ _
IDEALS _ _ _ _ _ _ _ _ _ _ _ _

A red-house is made of red bricks, has a red wooden door and a red roof. A yellow-house is made of yellow bricks, has a yellow wooden door and a yellow roof. What is a green-house made of?

JAJWUTH
TFAPOW
JFDABHC

What are the next five letters in this sequence?
Hint: Think back to childhood rhymes.