- planning of mine, the student and the company
- planning of mine, the student's, and the company's
Which is correct/better, and why? I would assume 2 is correct, but is 1 incorrect?
2 is correct and 1 is incorrect. This is because without the possessive -'s, there is nothing to show that student and company are possessive.
However, even 2 isn't written very well. It's standard to place yourself last in a list, and it looks a little clunky to have more possessive endings in a row than you can help:
Note that planning of mine would be completely fine (double genitive, as you noted); it's simply that the idiomatic double genitive is not common when the possessor is not represented by a pronoun (e.g. planning of the student's), and even less common in lists (e.g. planning of the student's, the parents', the school's...).
1 is certainly wrong, as @Daniel has explained. But 2 is confusing enough to be dubious, though not quite wrong. Far better to avoid repeated possessives entirely, as in "...planning by myself, the student and the company."