Generally 'feliz' and 'alegre' are synonyms. Though, I use 'cheerful' to mean 'alegre' in Spanish, because I could say 'Mi amiga es una persona alegre' (My friend is a cheerful person), it's not a temporary situation, she's like that; but 'Mi amiga es una persona feliz' (My friend is a happy person) probably because she's having success with her life. For me to be happy is a process, to be a cheerful person is a way of being. P.S. This is my way to make a difference between them. I'm not sure if it's okay though. :)

Both are correct :) Who told you that it's wrong??

Both are correct. You can say "Soy feliz" It means I am a happy person (all the time) "Estoy feliz" means I am happy at this moment.

私は幸せです! 私は嬉しい It's the same or not? en español "estoy feliz" "estoy alegre"