Yes they do stain your teeths , but usually its just surface stains . Just a simple polishing by your dentist will be sufficient to remove them .

Yes its true. A trick I like to do is to drink some plain water after done with coffee so at least the stains will not build up too fast.

If it is stained due to external substance such as coffee you'll need to go for scaling & polishing, then whitening depending on your teeth's condition