Is this scene appropriate to include in a story?

I can’t focus on all you’ve written because of lack of paragraphs. First you need to put some serious SPACE into your writing. *
Remember…Four (4 ) COMPLETE sentences make up a PARAGRAPH!
First things first, segregate your writing into easily read paragraphs.

It’s hard to know out of the context of the whole story. What age group is this directed toward? You also have some formatting issues. Also, never start a sentence with the word “And,” when you are writing formally.

When they got to school Peter opened the door for Rachel and let her through.
‘Thank you, Mr Bumby,’ she said. ‘I’ll see you in class.’
‘Goodbye Miss Pearson.’ Peter was feeling quite elated. He was glad Rachel had noticed his new grey trousers, and liked them. And he was glad she had called him Mr Bumby at last. This too made him feel grown-up. He had been preoccupied with dressing himself this morning, and his nervousness over the meeting with Rachel had rather unsettled his stomach.
As he walked into the toilets he met two of his friends. One of them, Frederick, was washing his hands at the sink.
‘Look at Bumby,’ he said, turning to George. ‘He’s all dressed up today.’
‘Mr Bumby,’ corrected Peter.
George looked round from the urinal. ‘Wow. Are you going somewhere, Mr Bumby?’ he asked.
‘I’m going to the toilet,’ said Peter as he entered a cubicle.
Both boys laughed as he shut the door. ‘He’s trying to impress Rachel Pearson,’ Peter heard Frederick say as he unfastened his belt.
‘Let’s go and tell her he’s on the bog, shall we?’ George taunted.
‘Shut up,’ said Peter as he sat down.
Further laughter followed as Frederick and George went out the door. Peter was thankful to be left in peace. They were right of course, he had wanted to impress Rachel by dressing up, because he liked her, and that was why he had called her Miss Pearson. He had been afraid afterwards that she would think him odd because he had called her that, and maybe she had at first. But she hadn’t spurned him, just continued to be friends. He hadn’t called her Miss Pearson again, but tried to avoid using her name. And now today, unexpectedly, she had addressed him formally, unprompted by him. Was it because of his smart appearance? Did it mean that she was returning his affections, that she liked him that way too?

There is a website where it is possible to post pieces of original literature and get useful feedback towards better writing; I’ve been a member for several years, it’s free and very useful. It’s called You Write On and by joining you agree to exchange critiques with other members – you can use a pseudonym – I recommend it.

It would depend on your audience. You really have to have an audience in mind to write to.

I don’t think the piece gave us any context for the story. So beyond being clueless as to the plot or pertinent details the dialogue seemed unpolished. However the feelings of uncertainty the mockery and so forth were relatable to real life people and situations so I don’t think you should scrap it completely. Just some revision would be helpful.

I suppose anything is OK to put into a story. The real question is, who is going to bother reading it and will they think it is any good. At that point, not so sure…

It’s highly unrealistic. Any second I was expecting one of these guys to ask to share the toilet paper. You need to do more research. You need to spend more time in the men’s toilet.