Everything that comes out of CERN is public, and is published in scientific journal papers so also readeable for non-CERN people (although of course I admit we use some jargon which can confuse non-experts).I am allowed to say anything I want, there are no limitations to that from CERN. If you want to read more I propose you start with some popular science books, for example the one by Jon Butterwhorth.

Essentially the whole list of 'theories' you give are fictional conspiracy theories and not true. We do have a statue of Shiva that was given to us by the Indian government when India joined CERN. Some of us don't like having a religious symbol on site but it is a pretty statue at least, better than having a large crucifix I guess. And it is just a bronze statue, nothing more than that :)

So:

• Other dimensions: no we are not opening portals. We do do research to microscopic extra dimensions as predicted by string theory. But haven't found any since starting to look for those in 2008 (so 10 years, no results)
• The logo is based on the accelerators on the CERN site, not on 666. Yes accelerators are round and are interconnected. Hence the lines and circles.
• The odd dance video: I think you mean the summer student prank? Some MSc students decided to make a fake conspiracy theory video. By the way they were severely disciplined, which I do not agree with, but the CERN management was definitely not amused.

So what IS the real purpose? We study collisions of one particle on one particle in extremely controlled conditions. Those conditions also happen in the earth's atmosphere, but are of course impossible to measure in detail there.
Why do we study those collisions: because we can use those collisions to figure out how all matter, atoms, etc around us sticks together and what building blocks it is made of. This is important knowledge that will help scientists figure out how the universe started to exist, but also gives us more knowledge on the world around us.