I've flown with airplane boxes. I have three boxes which meet the 62" requirement. Here are some pictures of the first one. The fuselage had to go in diagonally. It's easier if the fuselage comes apart somewhere in the middle. Wings go in the hot-wired foam piece on top. The green ribbons allow lifting of the wing holder. The box for my new dog has layers of 2" or 3" foam the width and length of the box with cutouts for the stuff. That works better. Don't forget the lifting straps.
Airlines worry more about space than weight. I figured I could have made my box out of .2" aluminum and met the weight limit. I used 5mm plywood with Kevlar on the inside. Airlines still did some damage to the corners, but not the airplanes. Traian Morosanu went to Europe twice last summer with a modular aluminum box. I forgot where he got the parts to make the box, but they look pretty cool.
My new box is 34" x 11.2" x 15.7". The old box is longer and not as deep, so's to accommodate the fuselage length and meet the 62" total.
Box has to be openable by TSA for inspection. Use a TSA lock. They can open it, but the general public can't. Inspection varies from place to place. The authorities have always let me be present when they inspected my box. In Seattle, they just swab the edges to check for explosives. In Chicago, the TSA guy took all the pieces out and wouldn't let me touch anything while he was doing it. I had a bunch of pictures and instructions on the lid, but he didn't look at them. Inspectors in Europe either X-rayed the box without opening it or let me take the pieces out to show them. In general, try to check your box through to your destination and avoid having it transfer between airplanes along the way.
Batteries have to go as carry-on. Here are the FAA rules:
https://www.faa.gov/hazmat/packsafe/more_info/?hazmat=7 . Tape the connectors. Taking batteries has never been a problem in the US. Igor had his batteries confiscated somewhere in the Middle East on the way to Australia.
Please come to the West Coast. We'll show you a good time. Big contests are the Northwest Regionals Memorial Day weekend in Roseburg, Oregon,
http://flyinglines.org/23.reg.flyer.pdf ; the Golden State stunt contest in October, probably near Sacramento; and the Las Vegas contest in November.