Fast and quickly are both adverbs, so could equally work, however 'fast' refers to running or vehicles, and to me, it would sound strange here, where the suggestion is possibly going more leisurely or walking...
"I am hurrying to the cafe" - is that not a valid translation, or not literal enough for DuoLingo? Nobody English would say "I rapidly go to the cafe" would they?
I can't really answer whether it's "not literal enough" - but consider:
- Mi rapidas = I hurry / I am hurrying
- Mi estas rapida = I am fast.
- Mi iras rapide = I'm going fast.
So, for this sentence, I would expect something like: - "Mi rapide iras al la kafejo." - I quickly go to the cafe / I go quickly to the cafe / I am going to the cafe quickly / etc.
Claims about free word order in Esperanto are often overstated. Other times they're carefully hedged and people miss the hedging words. I would consider your word order here "slightly unconventional but not quite wrong."
What is the word order for a sentence with adverbs for it to be conventional and definitely correct? Are they always best before the verb or does it vary depending on the overall sentence meaning and emphasis? Is it ever grammatically correct to use the adverb at a different point, ie. after the verb or at the end of the sentence as seen occassionally in english, and if so what are the rules around that usage?