Translation:My older sister is busy.
Although 姐姐 means "older sister", in this sentence it would be legit to translate it as sister only, as that sounds natural in English. Of course, the information that the sister is older is lost in translation then. But especially when translating a novel where someone talks about their sister, you would in most cases use only "sister" in English instead of "older sister" to make it sound more natural.
On the other hand, when someone asks: "你有姐姐吗？", they specifically ask for older sisters, and then it should be translated in all cases.
Nature is right, family terms can be very specific even when referring to mother's side vs father's side of the family. And people address each other (even those not related) "older sister" "older brother" "uncle" "aunt" etc out of respect, so it is VERY important you know the difference. It is a VERY commonly used thing for chinese culture.
是 is specifically used for equality, eg "She is my older sister", not for descriptive "is" sentences (noun is adjective). In that case, you need another word, usually 很 if no other adverb-ish word is appropriate.
As an example of another word to use, 她很忙 could be negated as 她不忙. I believe 她最忙 would be the correct way to say that she is the busiest as another example.
Because in Chinese they don't use "Is" for something that already is something. For example, You don't say " She is very pretty" If she's already to be pretty! instead, you say Tā hěn piàoliang 她很漂亮 literal translation means "She very pretty". I know for us that makes no sense, but that's just how Chinese works.
There isn't a plural form of the term. You'd just say the number of older or younger sisters you have followed by the classifier for the number of people followed by the older or younger sister term. This is the same with all nouns in Chinese. You'd just say the number of things followed by the classifier for that thing followed by the term for that thing.
I think I see what you're trying to say, but in Chinese, but it's inherently made known within the term whether the sister is older or younger. While just sister could be accepted, it's important to understand that the younger/older meaning is inherent within the term, in order to learn how to properly use the term. It might not make a difference to you, but I guarantee that you'd briefly confuse a native speaker if the wrong term was used.