1. After three days he was arrested and imprisoned.
2. Three days later he was arrested and imprisoned.
3. After three days, he was arrested and imprisoned.
4. Three days later, he was arrested and imprisoned.
[Can we use all the expressions? Do we have to use a comma?]
5. He will be back home after three days.
6. He will be back home in three days.
7. He will be back home three days later.
[Can we use all the three expressions? Which one is the most suitable of the three?]
3 and 4 are right. Yes, you need the comma.
5, 6, and 7 are right. Because the adverbial phrases are last in those sentences, no commas are needed in them. I think 5 has the best phrasing.