Provided you have a WiFi connection (or access to a hotspot somewhere), yes. You can trade online, as well as battling. You need to have the other persons' Friend Code and register them in your "Pal Pad" (and they need to have yours and register you), there's no browser that you can just search for people to fight. But as long as you register one another, you can freely get online and trade or battle.
G/S/C are incompatible, as they were with R/S/E. You'll never get your GSC pokemon into a current Pokemon game. They used completely different systems to deal with stats, so they'd have to allow a big overhaul for that to ever happen. You can consider RBY/GSC and RSE/DP to be completely different "sets" of games, in a way.
In the case of R/S/E (and FR/LG), once you get the National Dex in D/P you can use the dual-slots on the DS (one for the GBA game, one for D/P) and "migrate" your pokemon into the new games using a single DS system. The transfer is 1-way, meaning they can never go back to R/S/E, and you are restricted to 6 pokemon per GBA game pack every 24 hours. They come into D/P exactly as they were in the GBA games, right down to the ball they were captured in and the item they were holding. Its great to be able to continue your teams on from RSE, and its also the only way to get legendary pokemon from past generations and starters from past generations.