How can I add copyright info (like website/Facebook Page) to a picture (in Lightroom or other software) and have it shown automatically on every photo uploaded on Facebook?

You would have to state in your contract with whomever is buying your photos that part of your terms is that they show copyright and links when using your photographs. Photo information like that does not automatically pre-populate. I would also encourage you to use a service like Digimarc, which for a small yearly fee allows you to watermark your images and then be able to track their usage online. Its a plugin that comes preinstalled with most major imaging applications and you just need to activate it. This will allow you to track and view all usage and you can make sure they provide the right copyright information. As noted already I would also use a visible watermark but be aware that it can easily be cropped out.

In Lightroom 4 or 5 you simply go to File> Export> Watermarking> Edit Watermarks and you can create your own in the box at the bottom in the bottom right