Can "Monsta" be a correct slang word or abbreviation for "Monster"?
Sounds good and cute to me but as a non-native speaker I don't understand if it's actually a word that natives may use (maybe kids?).
Yes, it appears that monsta is used (albeit very rarely) in place of monster, similar to how gangsta is used instead of gangster.
According to the OED (← free page!), the spelling of gangsta is "intended to reflect an African-American pronunciation", and that's what I've seen for most of the examples I've found for monsta as well.