AT&T Text Messages - Computer

I just starting using the site on my computer to text.  A few questions...

Why doesn't it carry over any of my text messages for a conversation I have had via text with another iPhone user? (I know you have to turn the imessage off)

If I delete a conversation of the messages app on my computer, will it delete the conservation from my phone?

How do you delete multiple messages using the messaging app on my computer?  I can only delete one at a time and I have hundreds.

When I respond to someone, my message I sent to them via the computer messaging app doesn't carry over to my phone.  The only messages that carry of to my phone are the responses.  How do I fix this?

I can only see the phone numbers when using the messaging program. How can add names to these numbers?

Thank you