When bathing just don’t “scrub” like you would your own hair in the shower. Just kinda smooth it downwards towards the ends. Rinse the same way. Just pouring over top and letting it rinse away.
Also, dilute the shampoo/conditioner using these bottles https://www.amazon.ca/Wilton-6-Ounce-Melting-Decorating-Squeeze/dp/B0000VMBF6/ref=mp_s_a_1_3?dchild=1&keywords=Plastic+Squeeze+Bottle&qid=1618958038&sr=8-3. It helps to spread it on and resist the urge to rub it in. I use this shampoo https://groomerspro.com/natures-specialties-sweet-passion-shampoo-dilutes-8-1/ as it is made to be diluted with water.
Look up Crate Games. Getting your pup to be happy and comfortable in the carrier will help a lot. A diaper may not hurt, or at the very least some pee pads in the crate to absorb any potential accidents. Most airports I’ve been in have a dog potty area inside security, so you should be able to give your pup breaks between flights if you have connections.
We usually fly from Florida to Washington once or twice a year with our coton in cabin, which is about a six hour flight and it’s always gone pretty smoothly for us. I have this carrier which measures 10.5x11x18 and it’s always worked well. Our coton is 10 pounds and has plenty of room to turn around and whatnot in it. If you have any other specific questions I’d be happy to answer them.