If you have a valid ground to argue you did not voluntarily relinquish your status, you could try to do it.

You can send them a letter when you reach your destination outside of the U.S.

If your wife has an EAD she is allowed to work in the U.S. regardless of what you are doing.

An applicant for extension of status can stay in the U.S. until there is a decision.

The time you were actually in H4 status may be recaptured.

You ask about a K4. Then you asked about an I-130. If you are filing for your own children and you are a U.S. citizen, then they may be US...

Resubmit them. There is not any other way to handle it.

You can apply. But since you must prove non-immigrant intent to get an F1 visa, your K1 petition may preclude you from doing so.

I doubt that this would be considered unauthorized employment.

You have the discretion to put whatever date you consider reasonable.