You can still play online multiplayer games. Smash Bros, Mario Kart 7, Monster Hunter Stories PvP etc. The whole Pokemon Bank also works. They only removed the option in eShop to purchase new games, you can still also download everything you bought before.

You don't have to put it in airplane mode. The 3ds can't update automatically; you have to manually go to the software update menu.

Are online services gone with the eShop? Last I checked most games still seemed to work online (this was like a year ago though)