1. There is random feature in Battle Tower that friend codes are not required for. I believe the same applies to Battle Subway(although I haven't played that yet).
http://bulbapedia.bulbagarden.net/wiki/Global_Battle_Union#Random_Matchups you may want to look into Global Battle Union as well.
2. People still do play online in 4th Gen just not to the extent as years ago, of course with the advent of BW.
3. If you trade over Wifi Club then yes. If you do it over GTS then no. There is also GTS Negotiations which is random trading. My recommendation though is trade with people on forums like here in Trade Corner and just exchange friend codes with them.