I would like to know if it makes any difference storing it "strings down" or on its back (in relation with the bag), as I have heard different opinions on how this affects the guitar neck.
Storing a guitar strings-down probably isn't the best idea, since putting weight on the strings will grind them against the frets, putting a tiny amount of wear on the strings and frets. It may also be putting the weight of not only the guitar but the case on the guitar's neck, depending on how your case is constructed.
I suggest storing the guitar in the case on its back or one of the sides.
I'm pretty certain we have a very similar question here, but I can't find it...
Anyway - consensus seems to be that your best bet is to store it on its side if laying it horizontally, or better yet: store it on a guitar stand, vertically.
Either of these positions minimise the stress on the neck, frets and strings.