No, "jack" and "sat" do not rhyme. "Jack" rhymes with words like "back" and "lack."
Corner
Little Jack HornerLittle Jack Horner sat in the cornerEating his Christmas pie,He put in his thumb and pulled out a plumAnd said "What a good boy am I!"
Humpty Dumpty is the character who sat among the cinders in the nursery rhyme.
Little Jack Horner is the nursery rhyme character known for putting in his thumb and pulling out a plum. He did this in the nursery rhyme "Little Jack Horner".
The cat sat on a rat
Yup.
Little Jack Horner is said to have eaten a Christmas pie, not specifically a fruit pie. The nursery rhyme goes, "Little Jack Horner sat in a corner, eating a Christmas pie."
In the rhyme Jack and Jill, I estimate Jack and Jill to be 5-7 years of age.
The rhyme scheme of "I do not like the sunshine" by Jack Prelutsky is AABBCCDDEEFF.
A plum. The full nursery rhyme follows: Little Jack Horner Sat in the corner, Eating a Christmas pie; He put in his thumb, And pulled out a plum, And said 'What a good boy am I!'
Humpty Dumpty sat on a wall.
Christmas pie