"I am not busy this month."
Uh, I think this sentence can mean either:
As for this month, [I am] not busy.
As for this month, this month is busy.
like we are doing a lot of things for this month so this month is a busy month .
は just indicates the topic of the sentence. XはYです doesn't automatically translate to "x is y". It sometimes depends on the context to know what the speaker means.
I recommend Kim Tae's Japanese Grammar Guide (www.guidetojapanese.org). It's explained in a great way there.
です is the auxiliary verb "to be". I.e. you use it when something or someone is 'something'. For example, "I am a teacher" = 先生（せんせい）です.
ます is not a separate verb (with the exception of 増す, "to increase"), but a polite verb ending that is stuck onto the conjugative stem (the renyoukei 連用形) of a verb. For example: いく= "to go", the conjugative stem is いき, and so the polite form is いきます. As a whole, this is just a regular verb and can be conjugated as such (e.g. into a past tense いきました).
おります is the polite form of おる; "to fold/break", but I'm guessing you meant あります / ある. This is "to be" in the existential sense. I.e. when you say something simply is (or isn't). For example: 本があります = there is a book / there are books.
Odd. A Dictionary of Basic Japanese Grammar lists nai as the plain negative form of aru.
And Under aru1, there is a note;
"Aru expresses the existence of or the possession of inanimate things including abstract things such as events or problems."
So, since I'd call 'being busy' an abstract thing like an event or problem, ある seems to be exactly the right verb to use, with ありません being the proper polite negative version to use...
If anything, the use of です with an い-adjective like ない is less correct as you're not supposed to use だ/であるwith them, and what is です but an contraction of であります?
haha, I see, I think it sounds rather cute, like trying to make a negation less rude, like ナイ！！………～です。
It's also accepted that です after い～ is not part of the grammatical structure, it is just a deferential artifact. That's why you wouldn't say ありませんです because the ～ません already accomplish that deferential tone.