Yes to all questions. You can check the car seat, gate check it, or bring it on board if the child has a paid ticket and the child uses the car seat. If checking, the bag is optional. TSA does allow larger quantities of liquid: "Formula, breast milk, juice in quantities greater than 3.4 ounces or 100 milliliters are allowed in carry-on baggage and do not need to fit within a quart-sized bag. Remove these items from your carry-on bag to be screened separately from the rest of your belongings. You do not need to travel with your child to bring breast milk." https://www.tsa.gov/travel/special-procedures/traveling-children --TheMiddleSeat
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