I've checked some online dictionaries and have not found it in some while it does exist in others and so wonder if it's a word or not and unsure how to be sure about such things? Given a word, how can one be sure it exists? Is there an official online dictionary that can be trusted?
No, there's no "official" source of English vocabulary. Generally, you're allowed to form adverbs by adding -ly to adjectives, so your first guess should be that mediocrely meaning "in a mediocre manner" is a word. Then check to see whether people use it like this from Larry McMurtry and the West: An Ambivalent Relationship by Mark Busby:
In high school I did a surprising number of things mediocrely. Among these were baseball, basketball, 4-H work, tennis, track, ready writing, editorial writing, extemporaneous speech, drama, declaiming, trombone playing, and debating.
The google reports almost 40K uses online. Even allowing for instances of debate over whether it's a word, that leaves enough to validate your first guess.
The Oxford English Dictionary has an entry for mediocrely since OED Third Edition, June 2001. http://www.oed.com/view/Entry/251632