1. The prepositional phrase, "for her birthday," is an adverbial phrase modifying buy. It tells WHY you're buying.
2. A "little" boy is a young boy -- generally under age 6 or so. Older boys don't like to be called "little."
3. This doesn't make any sense. A sandal is a kind of shoe.
(Broken Link Removed)
A sleeper has several meanings -- but it's often a pair of pajamas with feet worn by young children.
4. "He gave me his heart" means that he loves me.
1. I will buy Mom a scarf for her birthday.
(What is the part of speech of 'for her birthday'? Is it an adjective phrase modifying 'scarf', or is it an adverbial phrase modifying 'buy'?
The prepositional phrase "for her birthday" contains a preposition ("for"), a possessive pronoun ("her") acting as an adjective, modifying the noun ("birthday") which is the object of the preposition. I'd say the prepositional phrase is an adjective phrase, modifying "scarf."
2. Look at the little boy.
(What is a little boy? A young boy or a short boy? Though he is a little boy, he is tall. Can we say like this?)
"A little boy" usually refers to his age, meaning a young boy. Yes, you can refer to a little boy as tall -- implying tall for his age.
3. Is sandle called sleepers?
Are sandlas called sleepers?
(Which one is correct? And is 'sandal' a kind of 'sleeper'?)
I think you mean "sandals" and "slippers." Both are types of footwear; sandals can be worn indoors and out; slippers are usually worn inside the house.
4. He gave me his heart.
(What is the meaning of this one? Do you have a hidden meaning in this expression?)
This expression implies that he loves me. In English, the heart is symbolic of love.