Learn Italian: Loro invitano tutti, eccetto il ragazzo allergico. - They invite everyone, except the allergic boy.

Yes! eccetto and tranne mean the exact same thing ("except") and are completely interchangeable. You could say ...tranne il ragazzo allergico and the meaning wouldn't change at all.

Do I need to put a preposition after eccetto, like a or per?

No, eccetto (just like tranne) connects directly to the noun or pronoun that follows it. You don't need to add extra prepositions. It is simply eccetto il ragazzo (except the boy).

If they were excluding a girl, would eccetto change its ending to match the feminine noun?

No, eccetto is invariable. Even though it looks like it ends in an -o like a masculine adjective, it acts as a preposition of exclusion here. It always stays eccetto whether the noun following it is masculine, feminine, singular, or plural.

Why does the sentence use tutti instead of tutto?

When used on its own as a pronoun referring to people, tutti means "everyone" or "everybody". If you used tutto here, it would mean they invite "everything".