"The student is not cool."
First, you have misspelled the words -- take note of the spelling of these words below:
멋있습니다 = "is awesome, cool, stylish"
멋있습니다 = "is not awesome, cool, stylish"
맛있습니다 = "is delicious"
맛없습니다 = "tastes bad"
멋 is a noun that means "stylish" while 맛 is a noun that means "flavor". The descriptive verbs above are formed by combining these nouns with the two adjectival existential verbs 있다 ("to exist") and 없다 ("to lack").
They use particles, and they are placed after the word. So, 은/는 is kind of like 'a', and 이/가 is kind of like 'the'. The two starting with ㅇ are used after consonants, and the RHS ones are used after vowels. The 'a' one is for a subject, and the 'the' one is used for a topic. So the 'the' one puts more emphasis on the preceeding word. e.g. Movies are interesting. <-- general statement about no particular movie. Use 은/는. Or The movie is interesting. Maybe you have already established the topic, so no need to emphasis the word movie. Hence, still use 은/는. But, if you want to emphasis 'movie', perhaps you are answering the question "What was interesting?" ... "The movie was interesting." Here, you use 이/가.
You can, but people would look at you strangely. When you negate a verb or adjective, the first preference is to use the antonym. The adjective here is 멋있다 and its antonym is 멋없다.
Etymology notes for the curious:
멋있다 comes from the contraction of the phrase 멋이 있다 = "has style". Its antonym is similarly formed.
Because of this contraction there are two correct ways to pronounce 멋있다:
- /머딛따/, thereby respecting the origin as two words or
- /머싣따/, thereby treating the contraction as a single word.
The second is definitely more common to hear however. This dual pronunciation is true for the antonym as well. But strangely, the preferred pronunciation is the opposite: 멋없다 as /머덥따/.