Yes, those forms are correct. There are many ways of talking about the future in English, and using 'will' is one of the ways. Your sentences don't sound very natural, but they are correct grammatically.

Yes, these are all grammatically correct. In American English, we would usually say "I am going to study five lessons today and six lessons tomorrow." though. "Will study" is generally used for making a strong promise and "going to study" doesn't convey any emotion behind it really. :)

You can say those sentences that way and they are correct, if that is what you actually mean. That is a lot of studying. If you are working on a series of lessons and you are doing one lesson a day, then you say it differently. Today I did the fifth lesson. Tomorrow I will do the sixth lesson.

must say: I'm going to finish five lessons today and six lessons tomorrow.