No. There are nearly two hundred sign languages in the world. Some are spoken by hundreds of thousands of people (such as ASL – American Sign Language, spoken in the United States and Canada), others are limited to small rural communities (like many local sign languages in Africa or Asia). Usually, deaf people in a country use the same sign language, which does not mean that it derives from the phonic language used in a given territory. British Sign Language, for example, has nothing to do with ASL, although hearing in the UK and the US uses the same phonic language.
Does sign language show letters with their hands? No. The signs of the signs do not refer to the words of the spoken language, they are not made up of letters, because these are, after all, the recording of sounds. Each sign corresponds to a single concept (eg HOME, WORK, NICE), so it resembles the words of the spoken language rather than the individual sounds / letters.
Do deaf people sign? The language of the Polish deaf community, usually called Polish Sign Language (PJM), is by no means a transmission of the Polish language spoken by means of signs. It has its own grammar. The question How old are you? in PJM it corresponds to the sequence of characters: You life, how much. Therefore, the order of the sentence is different, the choice of words / characters is different. For at least two hundred years (ie since the establishment of the first school for deaf children in Poland), PJM has developed and evolved spontaneously. Today it is a fully developed natural language which, due to its total uniqueness and diversity from Polish, has been ignored by hearing people for many decades.