"The school is not near."
ない is the negative suffix for adjectives and nouns, possibly others (definitely not verbs)
No. To make an i adjective negative you take off the i and add kunai. You do not change the desu (unless omitting it entirely to make the sentence less formal)
You can also remove the "i", add "ku", and then add "arimasen" afterwards
It kinda is. You can also think of it as a "not" when it comes to i adjectives.
I just came back from an english course for japanese speakers and it was scary. It was all kanji... They pronounced cat as see-at I didnt know how to get back, i was opening and closing every tabe for 10 minutes
ちかくありません is apparently correct, although it was not what I learned in my (admittedly very basic) course. As for ちかくないのです, there is no reason to add the の -- the correct form is ちかくないです