What are some problems facing students and teachers learning/teaching English as a second language which might be suitable for a research paper?

Here, I am assuming that you are referring to Teaching English as a Second Language (often abbreviated as TESOL). 

Research papers in this subject tend to be narrowly focused studies emphasizing a particular group of language learners or a particular pedagogical approach; they rarely address multiple problems at a range of levels. Although a PhD thesis or a book might cover multiple stages in English learning, a research paper would either focus on a single...

Here, I am assuming that you are referring to Teaching English as a Second Language (often abbreviated as TESOL). 

Research papers in this subject tend to be narrowly focused studies emphasizing a particular group of language learners or a particular pedagogical approach; they rarely address multiple problems at a range of levels. Although a PhD thesis or a book might cover multiple stages in English learning, a research paper would either focus on a single learning stage or compare and contrast how to teach a very narrow skill to students at different stages in their learning. For example, you might compare errors in article usage in first and second year Chinese students studying English in one particular program. 

Another possibility might be to look at how to apply different pedagogical techniques to teach the same material to students at different ages or learning stages. For example, you might look at different methods for teaching a single irregular verb such as "to be" to native speakers of German beginning English in secondary school versus ones starting English studies at the university level. 

Your key to a successful paper is to choose a very narrow topic which you can examine in great detail rather than trying to cover a large amount of material in a more superficial manner.