When using the Verizon Messages App, will a record of messages still appear on the bill/usage?
Yes. I use Verizon Messages with my Android phone and I see a log of all my text messages.
The total will appear on the bill. You can see the numbers, times, etc. via My Verizon online. When using iMessage this will not appear on My Verizon so to answer your question it will show online.