Hi Fragline.
The only people who have the power to delete threads are moderators, of which I myself am one. But we only delete threads when there's a reason to do so like if it's spam, a duplicate of an already existing thread, or we've been requested by the person who started the thread themselves.
As to why you can't find your fic... Well, you know that this is a forum... And threads move up and down according to the recent activity they receive. If you post a thread now, that'd become Thread #1 and would be the first in line. If I post another after you, mine would be Thread #2 and would push yours down to second in chronological order. But if a reply gets posted to your thread after mine was posted, yours moves back up to the top above mine. With that in mind, threads can get pushed all the way off of the current page you're looking at for any given forum section (known as Page 1). What you have to do is one of several things.
1 Check your member profile and click on the "Show All Statistics" link beneath your total number of posts. Then click on the "Find all threads" link which should display the link for your fic.
2 Click on the Page 2 link in the appropriate section and see if you can find your fic.
3 Click on the "Search" link at the top of any forum page and search for your fic by its title.
4 Go to the Fanfic section. Move all the way to the bottom where it says "Show all threads". Open that combobox and select the option "From the Beginning". Then you'll get all the threads ever posted in that section and can search more thoroughly if time-consumingly if your fic doesn't show up on the Fanfic section's Page 2.
Hope these suggestions help and have a good day.