It actually depends on the region you live in China. People live in the northeast part of China are more likely to use 都. But for people living in the south, it's less frequently for them to use 都 in this way.

Both with 都 or without are right. It really depends on the context. Here I want to talk about the case with 都. When we ask that question, there are two situations: 1) I would be implying: what sports do you EVEN like? I might be talking to someone who's really not into sports. I might be asking him/her to go play some sport with me but got a 'no' no matter what sport I suggested. Possibly with a big frown. 2) I assume that there are more than one sport this person likes. This sentence can be translated as: What are ALL the sports you like? But then it would be more appropriate to say: 你都喜欢哪些体育项目？Using 些 in stead of 种 to assume there are more than one answer.

I believe it is used for emphasis. 都 is an adverbial particle that means “all, entirely, or even." Without it, the sentence would still be grammatically correct, but there would be a slight change in nuance.

This is a good question. I had never realized this point till you pointed it out. "都" in this context express a verbal and casual feeling, while asking for a multiple item answer. You should not use it when you ask someone who expect your "look up to" attitude. Without it, the sentence is correct too. Besides, as Lee mentioned, "都" can be used to highlight a result that gave you surprise. A:“他是谁？” B:"他是新来的经理，已经来了一个星期了。“ A:”啊！我都不知道。"

"都"is not necessary in this sentence. Without 都is more correct grammatically.