Actually,it does not matter for lose "zai".But it is better to use "zai"wth our oral language.

First, the Pinyin of "中文" needn't be capitalized. It should be "zhōngwén" because it is just an ordinary noun in Chinese, I guess. Second, the word "一直" should be read as "yìzhí" rather than "yīzhí". The character "一" can be read as yī, yí and yì. The high-level tone "yī" only shows when "一" is used as a cardinal or a ordinal. Third, your first sentence "我一直在学习中文。" is grammatically correct. However, this sentence can express two meanings: "I am learning Chinese now and I didn't do other things since I started learning." or "I have been learning Chinese for a period of time and maybe now I am not learning." So maybe you need some extra adverbs of time. Your second sentence "我一直学习中文。" is also grammatically correct. But it lacks for something to tell others what's going on so it makes people confused. You can be understood but native Chinese speakers won't say it like this. You need to add something to express a complete information. For example, you may say "我一直学习中文，却毫无进步。" (Wǒ yìzhí xuéxí zhōngwén, què háowú jìnbù., I have been learning Chinese for a long time but it didn't seem to be improved at all.)

better put the 'zai' in there. it also makes sense without zai. zai indicates the tense, you are doing something.

Well, in my opinion, the first sentence with "zai" suggests it's a short-period action, like you have been studying Chinese for 2 hours; But the second suggests that you have been studying Chinese for maybe several years. However, I think in many cases, they don't have much difference, and no matter you add "zai" or not, you can be understood.

it should be "Wǒ yīzhí zài xuéxí Zhōngwén". The second sentence seems lose some parts in it, i think the "zài " is a verb in this sentence, if you don't add it that it is not a complete sentence. Just my opinion.