On the day a cadet is dropped off for basic training:
Does the Academy fly and transport them to the Academy?
If that's what you want them to do, then yes. They will buy the appointee an airline ticket and provide transport to the academy. In this case, you would want to check with the academy and organizations like the AOG to arrange to arrive the day before and spend the night.
If the parents wish to attend, what time that day does the cadet have to report?
In-processing starts, I believe, at 9am. It could be a little earlier. It continues with waves of appointees in groups 1 right after another. I.e. everyone is in line and they take groups of say 25-50. As long as you are in the line, you are good. I believe the line continues on until about 11:00 am. If your appointee is ALSO an recruited athlete, most times the sport will coordinate with you to all meet together and have a little get together prior to inprocessing. So the appointees can meet the other team mates. Then all go an inprocess together. Still go in the same long line, just together.
What clothing or supplies will the cadet need?
This is very easy. Only bring the clothes on your back. If you had to spend the night, you might have the one extra pair of underwear or change of clothes. Then, just bring personal items; e.g. deoderant, shampoo, tooth paste. And ALL your paper work. And boots if you bought them early. Try not to have more than say a back pack. Once you are there, they will take that back pack away from you and you won't get to see it again for 6 or more weeks.
Can the cadet contact their parents during basic training?
Yes, but ONLY VIA SNAIL MAIL. The U.S. Postal service. No email. No phone calls. Recommend you bring a couple of post cards with you. Already stamped and addressed. Then just fill them out and send them. When parents/friends/etc... write you, have them include a post card that is already stamped and addressed. It makes sending a note much easier for the appointee. They don't have a lot of time.