However, without copyright we wouldn't have free software
We wouldn't have free software licenses. But we would still have open, collaborative software development.
or huge investments in proprietary software,
Most software doesn't need copyright in order to be profitable. That's why so many software companies are moving from the software-as-a-product to the software-as-a-service model.
without patents all inventions would be secret
Humans aren't good at keeping secrets. All secrets leak sooner or later as soon as >100 people know about them.
Almost all big inventions require the collective effort of >100 people.
without trademarks there would be no high quality brands due to market saturation
My smartphone is an HTC. Most of my non-techie friends don't recognize that brand. The HTC's quality is awesome, and superior to the iphone IMO. It needs to be, because the only reason people buy HTCs is because of their superior quality for the same price, not because of brand recognition.
Also, brands will lose their meaning a decade from now when the semantic web really takes off. People will no longer purchase stuff based on packaging, but content.
So which information should be protected by regulation and/or treated as property, and why?
Information cannot be protected by regulation from third parties. It can only be protected by personal measures.
It's not a question what information "should" do. It's a question of what information will
do, and how we can adapt to that inevitability.