Firstly, "내일 학교에 갈 거예요" is the correct sentence. Secondly, 에요 is actually supposed to be written as 이에요. Both of these are semi-polite versions of, 이다. 이다 means "I am, they are, it is, etc." The difference between the two are when you're supposed to use them. 이에요 is supposed to follow something ending in a consonant. Like in the sentence, "저는(I) 미국(american) 사람(person)이에요(am). 사람 ends with a ㅁ, which is a consonant. That's why it uses 이에요. 예요 follows after words that end in a vowel, like in the sentence, "저는(I) 여자(woman)예요(am)." 여자 ends with ㅏ, which is a vowel. That is why it uses 예요. :]

There's actually no difference except for the fact that 거예요 is correct and 거에요 is not. This because the conjugation of the verb-like adjective 이다 becomes ~예요 when placed after VOWELS while it becomes ~이에요 when it's placed after CONSONANTS, and in this case, 거 is our noun, and since it ends in a vowel, we put ~예요 after it. If we use the unshortened word for it, 것, then that's when we use ~이에요 because this one ends with a consonant, just like in your examples but with a different version of the word. 나는 학교에 갈 거예요. = 나는 학교에 갈 것이에요. The "~에요" ending is usually used by a lot of Koreans through text on the internet and chats because it's understood that they mean ~예요, but this is not actually correct. 굿럭~^^

(으)ㄹ 거예요 is the correct spelling. However, it is fairly commonly misspelled by natives as 거에요 due to its pronunciation. Looking at the origins of this grammar structure, we see it actually comes from (으)ㄹ 것이다. 것이다 -> 거이다 -> 거예요 It means "is a thing". And that's how you express the future :)