does the order of words matter?
for example, do the sentences: "mi ne estas viro" and "mi estas ne viro" are both correct? and do the adjectives go before the noun like english or after like italian?
The first sentence is fine.
The second sounds incomplete - you would use it if you want to contrast, e.g. "Mi estas ne viro, sed virino" (I am not a man but a woman). So you would use it when you say what you are not and then add what you are.
Word order is not completely free. Especially things such as "ne" (not) and "ankaŭ" (also) change the focus of the sentence depending on where they are placed, since they can modify different parts of the sentence.
Adjectives can come before or after the noun, and sometimes both at once: “La arbo verda / la verda arbo / la granda arbo verda.” I’m grateful for the work that Duolingo does, but sometimes I feel there is not emphasis on word order. Esperanto word order does not and should not always mirror English word order (subject-verb-object): “Al la viro mi donis monon / Monon mi donis al la viro / Al la viro monon mi donis / Mi donis monon al la viro.” All of these sentences have the same meaning. This is part of the fun of Esperanto! The accusative frees us from SVO word order, how wonderful!
The accustive only frees us from wondering what is the object of the verb. Each of thy sentences above means "I gave the money to the man" but some of them emphasize a different part of the sentence: Al la viro mi donis monon would be something like "the man is who I gave the money to (and not the woman)" Monon mi donis al la viro would be "it was MONEY I gave to the man (and not a snake)" etc.