Transit through Abu Dhabi on Etihad

Yes, the carrier will protect you onto the next available flight. (I assume this was all booked on one ticket—that is to say, you paid for all flights at one time in a single transaction to form a single contract with the company.)
Again the carrier will accommodate you at their expense. Long delays at AUH pre-clearance were common until recently, so you are advised to proceed there with haste.
Your baggage is "screened" by US Customs at AUH. Usually they just show you a photograph of the bag on a computer monitor and you confirm that it is your baggage. If US Customs have any suspicions they will pull the bag from the system for inspection but otherwise that's it. The baggage is through-checked to your final destination in the United States.
No, the same rules apply.

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