First thing to check is the encryption, a 'new' router would be using WPA encryption, but a DS Lite only supports WEP. You'd have to log into the router's settings and change it.
$500
no
If it does it like 500 times! Oh, quick tip. Don't say it froze!
yes
On the Game card.
yes
yes
no the dsi has a different charger
gameboy sp or original
At any Game Stop or Best Buy
At a Gamestop or EBGames.
yes it can =)