1. I believe her to be very careful.
(Can we omit 'to be' in this sentence?)
You'd need to keep "to be" in that sentence. Another way to phrase it is this: I believe she is very careful.
2. It's already summer. Leaves are dark green.
2-1. It's already summer. Leaves have dark green.
(Which one is correct?)
2 is correct. 2-1 is not.
3. He kept her delicious cake.
3-1. The game kept me excited.
(Are both graammatical?)
4. Hey, listen! I wrote this song.
Really? Oh, it sounds/looks great.
(Are both, sounds and looks, correct?)
The first line is fine; the second one is good, too, if you use "sounds."
5. I asked they play at night, but they didn't listen.
(Is this grammatical? Check errors, please.)
No, it's is not. You should write this: I asked them to play at night, but they didn't listen.
6. Next morning she wore the dress to school.
(What is the part of speech of 'to school?' What does 'to school' modify?)
The word "to" is a preposition here, and "school" is the object of the preposition. Together they make a prepositional phrase, and the phrase modifies the verb "wore" since it tells WHERE she wore the dress.
7. Her father sent some books for her from London.
7-1. Her father sent some books for her in London.
(Which preposition is correct?)
7 is correct. 7-1 is not.
8. Her friend Youri made a bear doll for her.
8-1 Her friend Youri made a teddy bear for her.
(Which one is correct? are both OK?)
Both are OK, but 8-1 is better.
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