I would like to obtain a transcript of text messages from the last 2-3 weeks from a particular number to mine. Will Verizon do this without a court order?
No they won't do it without a court order and when you get that court order you'll be lucky to get about 10 days worth of transcripts.
Depending on which phone model you have, you can use an application to create a transcript from the messages on your phone.
Thanks. I have a Galaxy S3, do you know if there is an app for that phone that would enable me to get deleted messages?
No, once you delete them they are gone unless you have backed them up in some form.
SMS Backup & Restore would be one such program which will back up your messages for future reference. It won't help with messages you have already deleted before performing the backup.