Can I put hard flooring over berber carpet?
No, that's not a very good idea for a few reasons. One is that if water is spilled and leaks through the cracks in the hardwood floor then the carpet will soak it up and do lots of bad stuff do the hardwood floor above it. 

And I'm not a concrete expert but I think concrete on top of carpet would not be a good idea either. And that would be way more work and money than just ripping up the old carpet.

There's plenty of ways to remove glued-down carpet. You can use boiling hot water and a big scraper. Or you can go to a big-box home improvement store and rent a power floor scraper. I know that Menard's has them.