Both are right. In my opinion, 这儿is oral Chinese. 这里is more formal and can be used in written Chinese.

They are the same. The difference is that 这儿 is oral, 这里 is written.

Very few people use 这里 in oral Mandarin here but then again I've only lived in the north. :P

Use whichever you like. 这儿 is not only used in Beijing or the area near it, though 这儿 is much more common than 这里 in Beijing. The main difference between them is that 这儿 sounds more informal than 这里. Therefore, it's not suitable to use 这儿 in your doctoral thesis, for example.

It's nothing about "correct". 这里 is general, and 这儿 is a casual way to say 这里。This shortened casual form is developed in 河北、北京，but 儿化音 is also a part of the modern standart chinese, so it's neither "incorrect" nor a "dialect". In southern, most people don't prefer to use 儿化音，because most of southern dialects don't have this phonem. Many people don't even pronouce 翘舌音（such as sh, zh, ch, r）。 So, you can say 这里 or 这儿 as you want to, they are both "correct". If you want to sound like local people, that an other question. ;)