And to be even more polite, you can say 'would' instead of 'will' and add 'please' at the end: Would you share your textbook with me, please?

The best answer would be: "Will you share your textbook with me?" "Will" is more polite than "Can" which sounds a bit demanding. "Can" is also not technically correct because of course they "can", they have the ability, but it is a question of whether they are willing.

Out of your answers it would be best to say "Can you share your textbook with me?" and "Can we share your textbook?" Though normally people wold say some thing like "I forgot my textbook, would you mind sharing yours with me?" It sounds better and much more polite. It's like the debate with "Can I got to the washroom?" and "May I go to the washroom?" Yes, you are capable of going to the washroom, but right now you are asking for permission to go to the washroom. So naturally you'd pick the second answer. I hope this makes sense.

Hi. You want to ask, "May I share your textbook, I forgot mine?" OR "May we share textbooks during this class, I forgot mine." Of course you don't have to use these particular sentences. Look at the following link to see when to use MAY/CAN.

I think the owner of the textbook should be the one who shares. "Can you share your textbook with me?" You wouldn't use "the" in this context.