Peter could try to prove himself. Instead of getting discouraged, he decides to work even harder. He takes on more challenges on his own to show the Avengers that they're wrong about him. He might go out of his way to solve problems that the Avengers couldn't, just to make a point.
He might also get angry. Peter has a strong sense of justice. If he feels that the Avengers are being unfair, he could stand up to them. He'll use his quick wit and his powers to show that he's not going to be pushed around easily.
One possible reason could be a misunderstanding. Maybe Peter accidentally gets involved in something that looks bad to the Avengers, like being in the wrong place at the wrong time during a security breach. So they misjudge him and are mean as a result.
Peter might be very defensive at first. He could argue back, trying to explain his actions. After all, he is a teenager with his own ideas and he may think that the Avengers don't fully understand his perspective.
Peter is likely to be in awe. He might be a bit nervous at first, but also extremely excited. He has looked up to the Avengers for a long time, so meeting them would be like a dream come true. He would probably be stuttering a bit as he tries to introduce himself and show his respect.
Peter usually blushes a lot. Since he's a shy and somewhat awkward teenager, his face turns bright red when the Avengers embarrass him. He might stutter while trying to defend himself or come up with an excuse.
Tony Stark might use his quick wit and technology. He could immediately scan Peter to make sure there's no physical danger causing the hyperventilation. Then he'd probably crack a joke to try and calm Peter down, like 'Kid, you breathe any faster and you'll fly away like a human jetpack.'
Peter might get a bit flustered at first. He's young and looks up to the Avengers, so being made fun of could make him blush or stammer. But then he usually bounces back with a witty comeback of his own. For example, he might say something like 'Well, at least I don't have an old - fashioned shield' to Steve if Steve teases him about his suit.
The Avengers would initially be in shock when they find out Peter is sick. Clint Barton would be the one to keep a calm head and start organizing things. Wanda Maximoff would try to use her powers to ease Peter's pain if possible. Vision would use his vast knowledge to try and find a solution. They would all be very concerned and do everything they can to make Peter better, from getting the best medical help to providing a safe and comfortable environment for him to recover.
The Avengers might react in different ways. Steve Rogers would probably be polite but firm in getting to know Peter's boyfriend. He'd want to ensure that he was a good influence on Peter. Thor, on the other hand, might be a bit more boisterous. He could either be really friendly and enthusiastic, or his loud and powerful presence might accidentally scare the boyfriend a bit.
They might be a bit surprised at first. Maybe Tony makes some jokes about British tea and bad teeth, but in a friendly way. Steve would probably be intrigued by Peter's different cultural background and might ask him about British history.
Natasha Romanoff could show her emotions in a more subtle way. She might look at Peter's old things and a small smile would cross her face as she remembers his innocence and determination. Thor, on the other hand, might become more determined to protect the world in Peter's honor, as he respected Peter's strength despite his young age.