I've found the answer, and the reason Google has become crazy, returning wrong URLs and de-indexing my webpages is not related to W3 Total Cache, nor to its settings, but to my robots.txt. Here's how it was until 5 minutes ago:
User-agent: *
Disallow: /forum
Disallow: /blog/wp-trackback.php
Disallow: /blog/wp-content/plugins/
Disallow: /channel.php
Disallow: /obrigado.html
User-agent: ia_archiver
Disallow: /
Without a blank line between User-agents, the "Disallow: /" was applying to * all, including googlebot. And here's how it is now:
User-agent: *
Disallow: /forum
Disallow: /blog/wp-trackback.php
Disallow: /blog/wp-content/plugins/
Disallow: /channel.php
Disallow: /obrigado.html
User-agent: ia_archiver
Disallow: /
I guess it's only a matter of time until all goes back to normal.