A 20x4 LCD should do you just fine and, depending on the 16x2 you have, might even use the same library and just require a minor change in the programme to suit.
As Floresta said, shipping depends on where you live. It may be slow if you live in a Third World country, but the problems are all at your end and will be bad no matter who ships it. The Chinese postal service would have to be one of the best in the world, I have no doubt they make sure of that, and I guess this is one reason why China is a rising economic superpower. I have had stuff in a week from Shenzhen and I get fidgety if it takes two. Hong Kong is much the same.
Being an architect, I can't help feeling that the Chinese group their mail order houses together in purpose-built high-rises with a big mail chute in the middle and its own post office branch on the ground floor. Nobody would get a corner office because that's where the cargo lifts are.