I would like to know this as well.

Well, ni describes that something is a part of, or within something. If you do not use "ni", you are essentially saying "Today's meal is using eggs", and that focuses the word "using", instead of containing. It's kind of impossible to translate what it means, but it implies that the meal itself is using the eggs.

Just imagine a chocolate fountain that's nothing but egg yolk.

The topic particle just sets the focus on something specific, so the meaning doesn't change much. For instance 今日は料理にたまごをつかいます answers the question of "what I'm doing today". 今日の料理に卵は使います answers the question of "whether or not I'm using eggs". But the resulting action is the same.

I did not put は and it's wrong. Why is it needed if it doesn't change the meaning?

Well, sort of. "gohan" is cooked rice primarily, but it occasionally means meal especially when coupled with a time of day. "r~ouri" is cuisine, cookery 食事, "shokuji" is what is the bona fide "meal" word.

Whuaaah I always learned the kanji 玉子 (たまご). Is 卵 also たまご？ When would you use which?

"ni" points to something being a part (in) something. "ha" implies that it's the topic of the sentence.

Well yes and no, the thing is duolingo wants translations instead of interpretations, and adding "will" would make it a future tense which really doesn't exist in Japanese.

You're not wrong, and it would sound more natural in english, but since we're translating you need to watch out for things like this

私は in almost all cases is a given and a pointless addition, the focus is actually something being used in a meal, that's why you need には, it's really hard to ascertain where the focus is in an english sentence, but it's usually near the middle in longer sentences and near the start in shorter phrases.

I just asked 3 and they disagree, what kind of japanese speakers do you know? Remember, these are translations, not interpretations. There's a big difference between the two.