1. I like the band because they sing with cheer. (Is this one correct?)

2. He always does exercise/exercises before breakfast. (Which one is correct, exercise or exercises?)

3. James weigh 60 kg. Andrew weigh 55 kg. So James is a little heavier than Andrew. (Is this one correct?)

4. Andy is smarter than any other boy in his class. (What does 'in his class' modify?)

5. No other boy in his class is smarter than Andy, (What does 'in his class' modify?)

6. Let's wash our hand for the meal.
6-1. Let's wash our hands before the meal is ready.
6-2. Let's wash our hands before we eat the meal. (Are the three expressions grammatical?)

7. I have to get up around sunrise to catch the train.

7-1. I have to get up at sunrise to catch the train.

7-2. I have to get up on sunrise to catch the train.

7-3. I have to get up in sunrise to catch the train.

(Which preposition do I have to use? Which one is correct?) From 7 to 7-3

8. Well, make a right turn at the bank.
8-1 Well, turn to the right at the bank.

(Are both the same?) From 8 to 8-1

9. He was rather angry, so he didn't want to have a seat next to her.

(Can we use 'have a chair' instead of 'have a seat'? Can we use 'take' instead of 'have'?)

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Thank you for your help. I have posted my questions. Would you answer them?

1. I like the band because they sing with cheer. (Is this one correct?)

correct

2. He always does exercise/exercises before breakfast. (Which one is correct, exercise or exercises?)
It would be more usual to write, "He always exercises before breakfast." You could also use "does his exercises" or "does his exercise..."

3. James weigh 60 kg. Andrew weigh 55 kg. So James is a little heavier than Andrew. (Is this one correct?)
"James" and "Andrew" are singular subjects in their sentences, so each verb needs to be "weighs" to match.

4. Andy is smarter than any other boy in his class. (What does 'in his class' modify?)
correct; "in his class" modifies "boy."

5. No other boy in his class is smarter than Andy, (What does 'in his class' modify?)
correct; same as above

6. Let's wash our hand for the meal.
6-1. Let's wash our hands before the meal is ready.
6-2. Let's wash our hands before we eat the meal. (Are the three expressions grammatical?)
6-1 and 6-2 are both correct; 6 is incorrect; you could also say, "Let's wash our hands before eating."

7. I have to get up around sunrise to catch the train.

7-1. I have to get up at sunrise to catch the train.

7-2. I have to get up on sunrise to catch the train.

7-3. I have to get up in sunrise to catch the train.

(Which preposition do I have to use? Which one is correct?) From 7 to 7-3
7 and 7-1 are correct; the others are incorrect. 7 indicates an approximate time; 7-1 indicates a more precise time.

8. Well, make a right turn at the bank.
8-1 Well, turn to the right at the bank.

(Are both the same?) From 8 to 8-1
Yes, they are the same, and they're both correct.

9. He was rather angry, so he didn't want to have a seat next to her.

(Can we use 'have a chair' instead of 'have a seat'? Can we use 'take' instead of 'have'?)
You'd use "have a chair" only if you are speaking about actual chairs; if you are referring to sitting down on a bench or in a train, though, you'd use "have a seat" since you wouldn't be referring to actual chairs. And yes, you can use "take" instead of "have." They're pretty much interchangeable in this idiom.