1.) It is indeed possible to trade with people around the world. there is a building that you can enter called the GTS (or Global Terminal if you're playing Platinum). Inside, you can search for Pokémon that people have put inside the GTS and put up for trade. You can search for the specific Pokémon, gender and level you require. The only downside to this is you must have at least seen the Pokémon to record in in your Pokédex to search for it.
Likewise, you can offer one of your Pokémon up for trade and choose what you want for it in return. Give it a few hours (you can even turn your game off overnight and check it when you wake up) and see if the trade has gone through.
2.) There is nothing story-wise really that requires you to play any of the Third Generation games (Ruby/Sapphire/Emerald) to enjoy the Fourth Generation games of Diamond, Pearl or Platinum. However, if you do decide to buy either HeartGold or SoulSilver when they are released (which are updated remakes of the original Gold and Silver versions) then I would reccomend playing FireRed or LeafGreen to help understand (although you say you have played all the originals, so it shouldn't be too difficult.)
3.) There are now a grand total of 493 Pokémon (extensive, I know), including the original 151/251. All Pokémon are available in some way in Pokémon Platinum, but some of the original Pokémon will require you to trade for them in the Trade forum here at PC, such as the original starter Pokémon or the Gold/Silver starters. You don't need to worry about the older Pokémon being removed from existence; they are still obtainable and are widey used/battled/traded every day.
Here is a list of all Pokémon available.
4.) There is a system where you can access your local Wi-Fi (either in a hotspot or with a wireless router in your home) downstairs in any Pokémon center. You will be given a key item called a PalPad when you first go down there, which allows you to store the friend codes of your friends, or even people on this forum (you can access the Trade and Battle forums here at PC for more info). You then head into Wi-Fi, and start trading or battling! You also need to give your friend code to those whose friend code you add so they can interact with you (much like the Wii, as you said), and you can obtain yours once you connect to a router or Wi-Fi hotspot, and reading your PalPad then selecting "My Friend Code" from the menu.
I hope this has answered your questions, but unfortunately I am going to have to lock this now.
*locks*