1) How many letters did Imam Hussain write in His lifetime and to whom?
2) How many letters did Imam Hussain write on the day of Ashura and to whom?
3) Where can I read the letters written by Imam Hussain in English. Please also share the Arabic source.

I don’t have a comprehensive list of all his letters. They have to be searched in books of hadith and books of history, and they are in Arabic (not sure if you read Arabic). But here are some letters translated to English:

1- https://www.al-islam.org/nafasul-mahmum-relating-heart-rending-tragedy-karbala-shaykh-abbas-qummi/imam-s-letter-noblemen

2- https://en.shafaqna.com/53572/imam-husayn-s-letter-karbala-brother-muhammad/

This letter is the shortest and was written on Ashura

3- https://www.sibtayn.com/en/index.php?option=com_content&view=article&id=5976:imam-husain-s-as-letter-to-muawiya,-the-emir-of-syria&catid=613&Itemid=701