When you put the capo on the first fret and then fret the string at the 14th, WHERE are you inserting the business card?
I just measured 4 different business cards, and their thickness varied between 0.010" (.25mm) and .016" (.41mm).
So, if you're inserting the business card around the 7th fret, and it's LIFTING the string to fit - you're looking at somewhere between .015" and .025" of neck relief... Which is a lot, in most cases. Generally, a good guitar with a good setup will end up somewhere between .001 and .010".
You say you think the truss rod is already tight. Well, it'll need to get tighter to remove that excessive relief...
You also say the action is "a bit high" for your taste. Despite fact it sounds like your guitar needs a neck reset (straight edge points below tip of saddle), I suspect removing that excessive relief will improve your action quite a bit. That, combined with some judicious nut slot filing (if your F barre chord is tough compared to a G or A barre chord) oughtta allow you to realize substantial improvement.
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