"I have been learning Chinese for a year. I'm still a beginner."
Subj. + Verb + 了 + Duration + 了 is the chinese contruction to express a duration of an ongoing action.
No that's not correct. At least one need to be used in spoken Chinese, but in written Chinese you need them both.
I don't think that your suggested sentence specifies that the year of studying Chinese has been this past year (up to and including the present). Your sentence could refer to a year of Chinese study that happened a decade ago, whereas the sentence structure in this lesson specifies the duration of an action that is still ongoing.
You should cut the sentence as "我，学中文，学了，一年了” Of course we say "I have been learning Chinese for a year." in English, but we do need 学了 to make it more natural. In other words, you can take 学中文 as object and 学了 as verb, so you can understand more easily.
I think your example is a fine Chinese sentence, but it does not specify (as the English does) that the year of studying Chinese is this last year, up to and including the present. Your sentence could just as easily refer to a year of study in college that happened a decade ago (and you're still a beginner because you haven't been studying in the intervening years).