If I would categorize the above expression, I will do like this 머무르다=머물다 vs. 묵다 vs. 체류하다. If you like to say that "I am staying here for 6 months", 머무르다=머물다 would be better proper that others. So 저는 한국에서 6개월간 머무르고 있어요( or 머물고 있어요) 묵다 is usually used when you want to empathize the fact that you are staying in a "accomodation" So, you can use 묵다 like this. "저는 저 호텔에 묵고 있어요"(I am staying in that hotel) 체류하다 is a formal and official adverb. It is mostly used in written Korean like books, newspapers. Oh It woule also be helpful to keep in mind that there are some expressions where 체류하다 have to be used, like 불법체류하다(meaning illegally staying) Have lots of fun in learing Korean.

있다 머무르다 머물다 체류하다 묵다 if you translate as "stay", it is fine. here is the difference that I can explain. 있다 : same as "be" verb. [Here it is] "여기 있어” 머무르다-머물다-묵다 : stay (no chinese character used) 체류하다 : stay (used chinese character "체류”) usually korean use “체류하다” officially, like airport. if someone say "저는 한국에 체류중입니다”, this sentence is official.