Why does AVAST never work when you sign up for a trial?
Perhaps not for you, but for others it is working.
Why is the only correspondence the one saying your non-existent trial is being converted to a paid version?
Likely because in your situation that is the only one needed.
Why is AVAST too ignorant to reply to emails?
avast is answering all tickets.
Why does AVAST have a nonsense telephone number?
avast has chosen to only use the ticket system. They don't have a public phone number for support.
Why does AVAST forum let me log in but not use it?
Complete BS. If it wouldn't allow you to use it, this post of yours wouldn't be here.

We do not list companies as this forum is not meant for that.
But I will give you one name.
avast !