Why do some of my texts say delivered or read under them? Why do some say nothing at all?

It is a timing issue. If texts were sent close in time to each other, like in an ongoing conversation, then the first text would be timestamped, and the others would not. The iPhone seems to need a specific period of time between the last text of a coversation and the first of the next conversation before it time stamps. Of course, if you get a text from another person, then it would react differently.

The delivered message is unique to iMessage. This just lets you know it was delivered through Apple's system. If it says Read, then the recipient has "Send Read Receipts" activated on their device.