Why when I downloaded AVG or Avast from their websites and tried to
open it through the Ubuntu Software Center - did I get the warning
'this package is of poor quality and could damage your computer'?

Are both AVG and Avast for Linux "bad" products (the windows versions
are considered very good)????

Thank you

You get that warning because they have not been tested by Ubuntu, by Ubuntu developers, so they have no way of knowing what damage they may do to your system. If you have to install a program, get it through synaptic package manager.

Next, why do you think you need anti-virus programs for a linux system? Only valid reason is to scan a mail server. Are you running one?

The design of linux does not allow a downloaded program to run, unless you, the user allows it. You have to go out of your way to set it executable. As long as you do not do that, you do not need an anti-virus program.

Ah- Now I understand. I am not sure what you mean by a 'mail server'-- I use
Yahoo - accessed through Thunderbird, Is Thunderbird considered a 'mail server'?

Thank you for your help.

ThunderBird is a mail client https://www.mozilla.org/en-GB/thunderbird/