1) ***Is it OK or incorrect to add 的 at the end of the question -- so, 你是几月几号的?
Yes, it is correct.
2) ***Can I change the order a little bit and say 你现在中国的生活习惯吗? <--- Would this still be correct?
Actually yes, sometimes we speak like this, but it's not standard and we won't write in this way. Anyway, I sugget you to regard it as a wrong sentence. I mean, if you are very fluent in Chinese, it's ok to say so and you'll sounds very casual. But if you are not, it only sounds you are making a grammar mistake.
3) I was taught that 名片 is the word for a "business card".
***Is it true, though, that in some parts of China, the word 片子 is also used? If so, in what region(s) do they say that?
I'm not sure if there are people use 片子 to replace 名片. Through my experience, 片子 is usually used to refer to
1.an X-ray film
2.movie or vedio (it doesn't have to be a porn, but it's also truth that many people who see the word 片子 would think of porn)
3.photo (basically only photography lovers would say it, the others just say 照片/相片)