The question is probably, "how can I tell if *my own* questions are 'research level', given the fact I am not an academic?"

I think a generally healthy attitude to have in this situation is that you should consider yourself pretty darned lucky (at this stage of your career) to hit on a question which could be considered "research level" (much less cutting edge!), and thus you should be a little bit skeptical towards your questions being "professional level" before you've gained more years of experience. I don't want to discourage you by saying this, but I do want to encourage a realistic attitude.

I feel even more confident saying this since your two questions have been about category theory, the area in which I mostly work. As you know category theory has been worked over by hundreds of very smart people over a period of about 70 years and concerns extremely general structures of essentially algebraic type. The thing about such generality is that your question -- especially if it concerns basic concepts such as cartesian closed categories -- is with great probability either one where people know how to construct counterexamples (because, after all, with great generality comes great flexibility), or if not, it's probably true for some natural reason that people have hit upon after 70 years of collective effort. (It's different from how it is with say number theory, where the integers are what they are and there is less built-in flexibility, so to speak -- there one can hit upon some question that no one knows how to answer with somewhat greater probability.)

Anyway, if you want to submit categorical questions to MO, you should definitely do a little research on them beforehand. Use Google, Wikipedia, the nLab, and of course MathOverflow and Mathematics StackExchange, just to name a few. If the structures you are asking about have been studied since the 60's -- and this is something you can probably find out through judicious use of Google -- the chances are high that answers have been around for a while. On the other hand, if you are talking about more esoteric matters such as differential cohomology in a cohesive $(\infty, 1)$-topos, then maybe not -- but there the bar is pretty high for knowing how even to formulate a sensible question. (I've seen some evidence of some young people overreaching in this respect, and it leaves a bad taste.)

I suppose that people might assign greater cachet or prestige to MO than to Math.SE, but you should consider using both, and use your judgment which is likely to be more appropriate. (Unfortunately, it is really M.SE that is "Overflowing" with questions these days, with many questions going unanswered, and not because they're "too hard" for M.SE.)