When your message box is full, people can't send you more messages. How about allowing them to send you messages by deleting the last message automatically when the new message arrives?

No...

Some people have a habit to store private messages for later viewing.

Basically.
It can also be abused by sending a message over and over so they receive nothing but junk messages if they happen to go away for a few days.

Basically his opinion.
Nope.