@MellyBee Your best bet is to visit your friendly neighborhood bike shop and ask them for help in boxing your bike using one of their discarded shipping boxes. They'll have all the axle protectors and know-how to fit everything in the box.
One trick I use when boxing my own bikes is to use foam pipe insulation and a bunch of zipties to keep everything together. Remove pedals, turn the handlebars/stem parallel to the top tube of the bike, remove the seatpost, and be sure to pad underneath the brake levers with foam or bubblewrap..
If you plan on traveling with your bike frequently, investing in a hard-shell bike box may be a smart move. Just be prepared for overweight charges from the airlines.
This is spot on. When I've reboxed bikes I'll often make some additional reinforcement from folded card board. It's also helpful to just see how a bike is packed in initially. I'm sure you can Google some pictures. Typically the front tire is removed and the handle bars. Thoses are then placed next to bike in the box. Don't leave any loose parts floating around. Everything should go into bages and be taped to the box or the bike (really the padding around the bike.
Make sure you have the tools you need to reassemble the bike!