You can see what phone numbers text was sent to or from, but without a court order you cannot get the actual content of the text messages. Even then it is usually limited to the last 10 days.
You can try My Verizon>My Device>Send and manage messages. Typically it syncs with your phone though, so if they are deleted from your phone, they won't be there. On a rare occasion I find that it hasn't done a proper sync with my device smoothly and may see an old thread.
This is true. Integrated messaging would save them to the cloud and I'm not sure if iMessage has any sort of save features.