Yes! I am a Hebrew speaker and I'm Jonah was the first translation I thought of—and was surprised to find that it worked. Just a note though, in real life most people named Jonah in Hebrew would put the emphasis on the first syllable. It's incorrect Hebrew, but this is normal in real life speech for this name (and many others), while for the bird the emphasis will always be on the second syllable.
For anyone who's thinking the same thing, Duolingo recommends https://www.memrise.com/course/1087087/hebrew-alef-bet-print.
I spent a week or so on it and it made a huge difference.
“The” works the same way as in English. As for the indefinite article, “a” or “an”, since Hebrew doesn’t use them, in a translation you have to insert them where they would normally occur in English. Therefore you need to say “I am a dove”. The only exception I can think of is for example in a song such as Helen Reddy’s I Am Woman Hear Me Roar.