Теория:

Study some information about questions with modal verbs.
Вопросы с модальными глаголами могут выражать приглашение, предложение, запрос на разрешение.
Asking for permission / Making requests

Can I/ Could I stay here?

May I / Might I make a suggestion?
 
Giving permission / Answering requests
 
Yes, you can. / Yes, of course (you can). / No, you can't.

Yes, you may. / Yes, of course (you may). / No, you may not. / I'd rather you didn't. / I'm afraid not, etc.
 
Making suggestions / invitations

Will you / Would you / Would you like to have dinner with me?

Shall we have dinner together?
 
Answering suggestions / invitations

I'd like to. / I'd love to. / Yes, all right. / I'm afraid  I can't. / I'd love to but I can't. / I'm sorry I can't.
 
Making offers

Shall I / we close teh window?
  
Can I / we bring sweets?
  
Would you like me to do the washing-up?
 
Answering offers

Yes, please. / No, thank you. / No, thanks.