Or the concept of the word is what's being communicated through signs, but not the word itself?
I think the Original Poster might be of the widely held belief that sign language translates 'actual' spoken language into signs. For example, one could envisage a system where the sentence she is at home would have four signs, one meaning she one meaning is, one meaning at and one meaning home. This would be a system similar to signed English for example, where the language being signed basically is English in a manually coded form. This could be thought of as a system similar to morse code. However, this is not what sign languages are.
Sign languages have their own distinct syntax and morphology and are full languages in their own right. British sign language for example is perhaps best characterised as an OSV language, where as American Sign Language is often described as an SVO language.
So the answer to the Original Poster's question is that spoken languages do not "have" sign languages. Rather people in different areas speak different sign languages that are distinct from each other as well as being distinct from the oral languages spoken in those areas (although, like other languages, sign languages borrow from other languages, including oral languages such as English).
Yes, and no.
There are many different sign lanuages in the world, and they have a tree of descent just as spoken languages do. But there is very little connection between the sign language and the spoken languages of a particular area
For example, ASL (American Sign Language) is said to be derived from French Sign Language: both are utterly different from BSL (British Sign Language), both in vocabulary and grammar.