We don't specialize on the school / application side of things, but I'll give you my 2 cents. You may also want to ask an admissions consultant.
Most schools do look at both the overall score and the subscores. Once you get each subscore to an "acceptable" level, any overall improvement after that can come from either area.
Where do you want to apply? For most of the top-10 schools, a Q44 would be a bit on the low side, yes. From what I've heard, those schools are looking for a 45+ (generally speaking - this is not a hard cut off, more of a guideline). I've heard admissions folks say that if someone has a strong quant background in other areas (undergrad degree, work responsibilities), then they're not as concerned about a lower GMAT quant score.
For a 10-20 school, on the other hand, a Q44 would probably be okay, and so you could try to crank up verbal to lift the overall score.
In general, I do think it's a good idea always to do some amount of studying on both sides (unless one side is already maxed out). You have more room for improvement on your lower side but it's also usually easier to improve your strength - so study both!
No percentiles are not an input into a score. They are an output. Your score is calculated and that score has a percentile associated with it.