What syntax is used in Esperanto?
I understand from the Duolingo course that Esperanto allows a lot of flexibility in the order of its words. (Ie, the subject, object and verb may come in any order and the same is true for nouns and their adjectives.) However, surely there must be a fixed generic syntax for the language; without one, many speakers will simply apply the syntax of their native language, perhaps resulting in multiple dialects of Esperanto (eg. English Esperanto, French Esperanto, Russian Esperanto, Chinese Esperanto etc.) rather than one universal language.
The use of the accusative case means that yes, people use the word order that they're used to, but others understand them anyway. Different ways of using the language are still understood, and there isn't X or Y dialect of Esperanto, because la kato mangxis la muson, la muson mangxis la kato, la muson la kato mangxis, etc etc, all mean the same thing.
There isn't a fixed syntax, as far as I'm aware, precisely so that learners don't need to learn the word order, but can use the language however is most comfortable and natural to them whilst still being understood.
SVO is, in my experience, typical, but that's mostly due, I think, to the influence of the native/known languages of the speakers. Other word orders are just as acceptable.
I'm not sure "syntax" is the correct word here. It sounds like you're asking mostly about word order.
"Dialects" of Esperanto in this sense generally don't exist.
My advice as a teacher of Esperanto is to focus on learning grammar and vocabulary now, copy the word order as presented in the sample sentences in the course, and when you are comfortable, start reading lots of good Esperanto (perhaps starting with the easy writings of Claude Piron). In this way you will absorb the rules naturally as you go.
The basic word order is SVO, adjectives before nouns, adverbs before the words that they modify, and other clauses at the end of the sentence.
These merely come from common use, not the rules of grammar, and so in theory the word order is quite free. However, they cannot always be broken without changing the meaning of the sentence. In sentences without direct objects, like "bovaĵo estas viando" ("beef is meat") the order matters, because "viando estas bovaĵo" would mean "meat is beef."
If you are at the point where you can read Esperanto, PMEG is a great resource for Esperanto grammar, including accepted practices for word order. http://bertilow.com/pmeg/gramatiko/vortordo/index.html