I study English
I study the English language
Is it grammatically correct to drop the:
I study English Language
The presentation will be made in French language.
No, it's not grammatically correct to drop the when English language is a standalone phrase like in your examples. And by a standalone phrase I mean a situation where English language is not being used as part of a bigger expression where it serves as an adjective. For instance:
I like visiting English language websites.
Since the arrival of the Internet, English language education has become more accessible.
Even if there is the, it goes with the noun that English language modifies, not English language itself:
I really enjoyed listening to the English language audio programs that you recommended.
Other than that, there should absolutely be a the in front of English language because English is a unique and very specific thing. Sure, there are different accents, regional dialects and what not, but historically and linguistically speaking, English in its totality is a thing that's one of a kind. There are not two or three different "Englishes" just like there are not two or three different "Frenches" or "Arabics". The English language is as specific and unique as the White House or the Eiffel Tower. I would even go so far as to say that saying English language without the sounds unnatural. So, it's always THE English language or THE French language:
How old is the English language compared to the French language?
However, you don't need the definite article when English is used on its own because in that case it's a proper noun—the name of the language (as you probably know, you typically don't use definite articles in front of proper nouns in English):
How similar are English and French?
The above is true unless you're talking about different dialects or types of English specific to a particular region, place or situation:
The English spoken in the Southern United States is very different from the English spoken in the state of New York.
— What exactly is it that you did not like about this book?
— I specifically did not like the English they use. It's very difficult to follow and contains a lot of grammatical errors.
I hope my explanation did not sound too confusing to you. If it did, let me know and I'll see what I can do.