Apple provides a one year limited manufacturer's warranty for the iPhone, and the AppleCare extended warranty extends Apple's standard one year warranty be an additional year. Both the standard one year warranty and the AppleCare extended warranty is a manufacturer's warranty only - it is not insurance so it does not cover accidental damage, or if the iPhone is lost or stolen.
No, it won't work and your dishonesty means the rest of us get punished because it forces Apple to become even stricter when legitimate warranty problems occur.
You dropped the phone and you need to take responsibility for it. You can call your credit card company and see if they cover accidental damge.
No cell phone manufacturer offers an insurance plan. Some cell companies will offer plans for some of their handsets but not all. You can google third party insurers and see what comes up.
Doubtful Apple is that stupid. If you call AppleCare to handle this over the phone, your iPhone will be picked up and returned for inspection and testing. If you make an appointment at an Apple store if there is one nearby, your iPhone will be inspected and tested - they won't just take your word for it and give you an exchange under warranty. This isn't Apple's first rodeo.
More than likely this problem will be determined to have been caused by accidental damage. You may get lucky by not be truthful up front but if you prefer being deceitful, then hopefully you won't reap what you sow. Even if you are truthful, you may get an exchange under warranty anyway.
Honesty is usually the best policy, so taking it to Apple telling them that your iPhone has a blank screen that just mysteriously appeared only is not being honest. Tell them you use a case - have your iPhone in this case if you make an appointment at an Apple store, and that you dropped your iPhone from hip level when this problem occurred. If there is no visual damage that indicates a drop, and nothing turns up in the testing that will be performed indicating this was caused by accidental damage, then you should receive an exchange under warranty.