If manga counts, I recommend trying to find magazines (monthly or weekly). These will have a variety of short chapters, so if one story is too advanced or uninteresting to you you can skip to the next.
Shoujo or shounen are the easiest, because they usually have furigana and easier grammar, but if you don’t mind looking up unfamiliar characters, etc., seinen or josei might be more interesting for you.
I managed to snag several bags’ worth of free monthly magazines from an anime convention this fall, and it’s been a lot of fun reading through them. I imagine you could find some online if you wanted to try something. Some suggestions are the Monthly Shounen Ace or Spirits Magazine (seinen).
Back to some of your other questions, I didn’t start reading Japanese much until the last few months (approx. level 17 before my reset), largely because my grammar is horrible. But with knowledge of some kanji and very basic grammar, most manga I have tried is readable, or at least understandable. Last week, I blustered through a short piece called Bokutachi no Koto, which is very lacking in dialogue and a seinen. I think I understood most of it, and if you have some reading experience you should have little trouble.
Really, you can start reading whatever you want as soon as you like–at your level, you should already know a good number of common characters and a bit of vocab. What to read is largely based on your tolerance for looking up what you don’t know.
Hope you find something that interests you!