Unless your (sole-member) LLC elects to be treated as an S-Corp or C-Corp it has no separate income tax filing requirement at the federal level.  It's considered a "disregarded" entity and you report its activities on Schedule C of your Form 1040.  While I expect most states take the same position as the federal government with respect to an "income tax return" for the LLC, there almost certainly are other annual state-level filing requirements so the state can extract an fee from you for the "privilege" of forming an LLC in their jurisdiction.

I don't know what you mean by "due date for filing start-up and organization costs."  Typically these costs are capitalized and then amortized over some period of time.

Tom Young