Shouldn't be a problem. Do bring your ID and valid immigration documents like you said just in case someone asks.
I take it your kids have valid travel documents as well (e.g. a US passport since they're US citizens) so they should have no trouble either.
No. If you have a valid employment visa, you may apply for dependent visas for your minor children. If they are approved, then you may go and pick up your kids with their visas.
Their visas expire at the same time yours does, and they must leave the country with you immediately upon completion or termination of the job. If they are in the middle of a school term, tough, they have to go. Consider whether you want this kind of upheaval in your children's lives for just a temporary work permit.
Remember there is no "grace period" to depart, and transportation for all of you is at your expense. Make sure you have the money saved up! Because if you were ever fired or laid off, the visas terminate that day, so you would have to leave immediately. Otherwise, the last day of work is the last day the visa is valid, and then you must leave immediately. Make sure you have the money saved up to take care of the transportation for all of you.