di does sound like “die,” but written down and in verbal contexts, i wouldn’t make the connection, and if i did, i would assume it was a name. if that worries you, then i think you could get away with pronouncing it “dee”, which is cute!
it’s not old ladyish to me at all!
Love it. Classic, but not overused.
Basically, Diana Barry. 
Sweet, kind, lovable, etc.
Charlotte because of the Royal family? Honestly, I think it would be fine. They are both such strong classics that I don’t think it would be too noticeable.
I’m not getting the Lily association, though… 
Hm… Personally, I don’t think Diana needs a nn, but…maybe Annie? (This is a tough one to think of nicknames for! 
Sorry I can’t be more help!)