You can bring your own beverages within reason. Bottled water is pricey but I didn't think it was $3.50. The only thing about packing drinks is think of the weight of lugging them around until you unpack.
As to your own thermos, yes you can fill it with iced water/ice tea from dispensers around the ship. Princess doesn't have lemonade available unfortunately.
RCI Sovereign of the Seas 2005
Caribbean Princess 2005
Carnival Paradise 2004