Can I open the window?
= Do I have your permission to open the window?
Will you permit me to open the window?
Is it all right (with you) if I open the window?
Do you mind if I open the window?
Could I open the window?
= Can I open the window?
Shall I open the window?
= Would you like me to open the window?
Do you want me to open the window?
(Also, Should I open the window?)
In these sentences, "can" and "could" are being used a substitutes for "may", i.e you are asking permission. "Shall" on the other hand is being used, not to ask permission, but to get another person's opinion
Note: In American English, shall is hardly used at all except as shown above in a first-person question, and even here, the paraphrases shown are often used instead. You can live your whole life in the U.S. without ever saying shall.
See this video for more information.
https://www.youtube.com/watch?v=68Fu8KZWb4Q
