Why do you not say "Ja jem jabłko i chleb"? Can you say it either way and it means the same thing?
Yes, you can drop pronouns in Polish. The form of the verb will tell you who's performing the action.
In most situations, it's more idiomatic to omit the pronoun. Both are correct.
Why is "an apple" and not just "apple"? Like in the bread case
The noun "apple" is countable, the noun "bread" is not.