стола is the genitive case of стол, as required after возле to mean 'near'.
Unfortunately, you are wrong on all your claims.
There is a Russian word for “to be”, it is «быть» (along with its conjugations «есть» and «суть»).
«Дверь возле стола» can be interpreted as a part of a sentence and there is no way to tell without context. Even if words and collocations begin with lowercase in this course, it is possible (as demonstrated by HatemShah) to have this phrase as an incomplete sentence on its own: “Which door to the office?” — ”The door near the table”. I don't know about English here, but in Russian it works fine.
«Дверь возле стола» can mean ”the door near the table” and not even under extreme circumstances, so I think this answer should be acceptable as well.
The Russian example suggested that the translation to English should be a complete sentence, indicated by the period at the end of the phrase.
The period doesn't indicate anything, incomplete sentences (in Russian) can end with periods too. The point was actually that it can be both a normal sentence:
Дверь (находится) -> (где?) возле стола.
...and a clause where the predicate is missing:
Дверь -> (которая?) возле стола.
the word «быть» exists, but Russians don't have word for the present tense "is."
Yes, we do. The present tense, 3rd person form of «быть» is «есть». There is a significant usage difference between Russian and English (and it is true that you often do not need it), but it still exists.
Look, I'm not trying to teach you English here. If you believe that saying ”the door near the table” while pointing at it is incorrect or colloquial or for some other reason unacceptable, your comment is not exactly super clear. Yes, I know that it is not a complete phrase, but so can be the sentence you were to translate.
ACTUALLY, I'm not incorrect on all my claims, what I was missing was to clarify the Russian usage of «быть» "to be." The Russian example suggested that the translation to English should be a complete sentence, indicated by the period at the end of the phrase. In ENGLISH (I am a native speaker and teach grammar), in order to form a COMPLETE sentence, you need a subject and a verb. The subject DOES something. "The door near the table" is ONLY a complex subject. Is there a verb? No, because "the door near the table" needs TO DO something. Ex. "The door near the table closes/IS closing." The door near the table opens/IS opening." Without a subject AND agreeing verb, it is not a sentence. So no, "The door near the table.<(PERIOD)" is not correct. It must be "The door IS near the table.(period)" in order to be a full functioning sentence IN ENGLISH because that is the target language to which we are translating. We are not reinterpreting the given Russian phrase.
Also, to address your commentary regarding use of "to be" in English and «быть» in Russian: the word «быть» exists, but Russians don't have word for the present tense "is." In English having "is" in a sentence with a period at the end is imperative.
Maybe you shouldn't support your position and arguments with "I don't know about English here..." and then proceed to give your adamant opinion on your clearly colloquial and incomplete understanding of English.
See here for a reference on prepositions and grammatical case: https://www.alphadictionary.com/rusgrammar/prepgen.html.