"I do not eat meat or fish."
In negation, the objects do not receive the action
it's not true though, you can also say 肉と魚を食べません、the は doesn't mean that the action is not being done, is simply stressing the negation of the action. It just happens that for general things like this, using は sounds more natural for Japanese people because the particle throws the focus into the negation.
a different example:
「私は『肉と魚を』食べませんが、ジョセさんは食べます」"I don't eat meat and fish, but Jose does"
In that case the は goes to 私 because I wanna make a contrast with "Jose". The action is still there with an object being marked. In fact, that assertion is conceptually wrong, because particles don't mark verbs, they mark non-verb things.
In the case of「肉と魚は食べません」は is superseding the を、that's all. The を is still there below は。
Yes, both are correct and generally you use を for positive and は for negation but the 'better' one to use also depends on the context (ex. the question that was asked before this). However, this is more complicated but for now we should probably stick to this.
sakana to niku wa tabemasen 魚と肉は食べません
sakana to niku wo tabemasu 魚と肉を食べます
Source: my Japanese friend.
Depends on the person. I know plenty of people who consider themselves vegetarian but eat fish or chicken or both - one friend claimed she was vegetarian but ate fish, chicken, bacon, sausages and mince (ground meat) - but wouldn't eat steak, chops etc. Yet others are very strict and won't so much as eat meat flavoured (chicken or beef) noodles or chips.
That's what I put as well and it wasn't accepted. I think the missing "ha" or "wo" between "魚も" and "食べません" might be part of it. In later lessons, they use "も" to signify "neither this nor that" in a negative sentence, so I can't see why it wouldn't be accepted. Next go around I'll add in the "wo" and see if it accepts it.
EDIT: Just checked, it doesn't accept にくもさかなもはたべません or にくもさかなもをたべません
I'm reporting one as "should be accepted". Out of context, it's hard to tell what the intent of the sentence is. If it's "I am not currently in the act of eating meat or fish" then "と" makes more sense. If it's "I do not typically eat meat or fish (because I'm a vegetarian or some other reason)" Then "も" would make more sense.
(If my understanding is wrong, please explain!)
I never learnt the grammar for it so I'm not entirely sure but after living for a while in Japan, I'd say も replaces the particles that should be used if the word it's attached to was not in an enumeration. Therefore you can not have もは and もを. Also, i haven't heard much of a double も. In other words, i think the sentence you're looking for is 肉と魚も食べません. But indeed, it feels like a better translation for "i eat neither meat nor fish" than of "i don't eat meat and fish". I don't think it has to do with what's the topic of the sentence.
About 10 years ago, at a Ryokan in parts remote, the booking had made clear that the group included several vegetarians. They got no meat - but twice as much fish as the non-vegetarians. The whole, deep fried flying fish (fins/wings outspread) were particularly special.
Of course, the proprietor may have just been an a-hole!
One I have run into in Japan is fish in "Dashi", a base cooking stock. High quality Dashi is perceived as being made from fish, not just seaweed. So if you go to a restaurant, and say "is your dashi made from fish", they might look insulted and reply "of course"
But if you say "I cannot eat fish or meat, what is OK?", you sometimes get a different answer!
Truth is they never explain anything, they teach you the same way a parent does.
If you started learning japanese here first, I'd say that's a bad idea, personally I used Wanikani and kaniwani for 2 months and bunpro for a month, and all kinds of little things in between before I came here and I'm glad that's how I did it, I view this platform as a place to practice, not really to learn.
(Note on sounding more natural)
I'm being a bit picky I suppose, but the choice to use "と" and "は" in the same sentence like that seems strange. The two main forms of "and" are "と" and "や." They're basically interchangeable except that と is implying these things specifically, while や implies "among other things." The difference between が and は is basically the same. が is emphasizing this/these specifically, and は is more general and open ended. So in most cases it has an unnatural feel to use the opposite implication in the same sentence.