I've been published a few times now, hoping to get 10 publications for the avatar within the year! I've never had a submission rejected, although I have had one held for several issues before it was published. They accept all types of articles - listicles, quizzes, column-styles, reports, opinion pieces, just about anything. I'm not sure about getting stories published, but I believe it's easiest to get one published when there aren't too many active series. It's a little harder to get published now, because the NT doesn't seem to come out consistently anymore, but don't let that discourage you! The plus there is that it doesn't seem to matter anymore when you submit, as all submissions appear to be considered at one time rather than singularly as they roll in.
When you have an article idea, search up the keywords relating to it in the Neopian Times search box. Make sure your article hasn't already been written! They aren't likely to publish an article on a specific topic that's already been written about, unless lots has changed since the previous article was published.
Ensure your article is written in the tone of Neopets culture. Articles shouldn't talk about Neopets as though it's an online game, but rather as though you are a citizen of Neopia, caring for your pets within that world, culture and lore.
Pay attention to your tone. Read out loud what you've written - if it feels awkward, it probably is. Try to write the words you would speak out loud if you were explaining your topic to someone verbally.
Proper grammar and spelling are a must. Run your article through an error catcher, or if you're hearing, try listening to it through Google translate by copying and pasting it into the translation box and hitting the speaker - I'm deaf, but I've been told a lot of hearing people find this to be a great way to notice their errors.
I hope this helps!