I think the omission of Level 2 grammar at JLPTstudy.com is because the author hasn't gotten around to it yet. It's a big task and as he comments on his site it can be very hard to find out exactly what the syllabus for JLPT is. (I'm glad he got the official Japanese book and made his site. It makes it easier for everyone else.)
There is quite a leap from JLPT3 to 2.
JLPT3's grammar fits in a nice little A5 book. JLPT2 from the same publisher seems to span 3 bigger volumes from what I can see.
Textbooks like Genki and Grammars attempt to do different things.
My experience is a textbook tries to get you using Japanese in many different ways in manageable thematic chunks. Grammars on the other hand are usually reference works. Best for expanding topics you've already encountered and checking points when writing.
Grammar is the structure and very useful but you'll find you need lots of other skills to use a language. Vocabulary being a big one. I'd start out with a good general text like Genki or Japanese for Busy People.
That said the series of grammar dictionaries are the best laid out grammars I've seen. Unfortunately they're a little expensive in the UK.
Kodansha publishes a range of
small books on Japanese grammar topics. I think Taeko Kamiya's 2 handbooks, one for verbs and one for adjectives and adverbs, are quite good. So are Naoko Chino's books on sentence patterns and particles. The whole series is well worth a look. There are books on particles, on onomatopoeia, on conjunctions, etc.
You sound like you are at the beginnings of your studies. I wholeheartedly agree with Psy-san about how language learning is an open-ended never ending task. (Indeed everything is. Cooking, Photography, Science, When you meet people who are good at what they do they are always learning, always curious, always enthusiastic. )
While having the target of fluency is good, I'd recommend you set yourself some more medium term waypoints like sitting JLPT4 before worrying about JLPT2; exchanging email before writing software manuals. And even these tasks you need to break down further and as you go along you need to review your goals and progress from time to time.
There is no Game Over unless you say so, that's what's good about learning.
頑張って〜♪