Yes you can access the Internet via your router, but you'll need to go and buy the Browser. (costs around £20 the last time I checked)
Yes, there are online games - Mario kart DS is probably the most popular.
For Wii/DS interaction - No official word as of yet but probably just a matter of time.
I've got the same games as you plus Yoshis Island so I cant really reccomend any games, I'm sure that others will though.
