I'm pretty sure that "salir' in this context usually has the connotation of 'going out' to party, on a date, etc. "Leaving," as if one went away to school every Sunday night, is usually expressed by 'irse," If I ask "Podrias salir conmigo?" I am asking for a date. "Would you go out with me?"
No, todos actually means all/every. In this case: all the sundays. But "you go out all sundays" is not what one would naturally say in English (but it is accepted by DL). So you should translate "I always go out" with "siempre", but you can translate "always on sunday" (literally: every sunday) with todos
I wouldn't agree to it being a more accurate translation. Your sentence conveys a similar meaning but I anticipate that it may have a very subtle difference in meaning. The answer we were given seems to be emphasing that you 'always go out' and your suggest emphasis on 'every' . IMO
"all the Sundays" would probably not be good English in most contexts-- although I can think of one where the "the" would be used.
In English, use "the" to refer to specific events, not to events in general as Spanish uses it.. for example, "the Sunday she left home.." refers to a specific Sunday. Whereas, "she goes out every Sunday" or " she goes out on Sundays would not have a "the." . These last two refer to "Sundays" in general.