Sorry, can't help with gothic contests. But there are a fair number of contests for 'speculative' fiction (which includes all manner of sci-fi, fantasy, horror, etc). The only upcoming one I know of is NZ-only, so it's of no use to you, but you might do a quick online search for speculative fiction contests.
This looks like a good challenge. I enjoy writing against a deadline - procrastination is otherwise a perennial vice and few times are as productive as the final 24 hours.
Whilst I don't write copper-bottomed 'Trekkie' type SciFi, would dystopian fiction qualify, do you think?
I expect it would be fine, provided there is some science involved. Do you read New Scientist (if you don't, you should)? It's quite a wide-ranging mix of science--biology, ecology, physics, astrophysics, chemistry, medical science--they cover it all. Even pseudoscience and silly, marginally-related-to-science topics, in a hilarious section at the back of every issue. So, yes, I expect them to accept a broad range of writing as SciFi.