Your second question is the easiest, WiiConnect 24 is the 24 hour on update service for your Wii, it allows the news and weather channels to update their info and for Nintendo to send you e-mails letting you know about firmware updates and other changes to system services.
As to the best way to connect, hmm. That's a tough one since it will vary from person to person. The dongle that Nintendo sells is supposed to be good in that it shares your PC's internet connection, getting around routers problems but you need to have a free USB port and be in range (I don't use one so I don't know what the effective range is). IMHO if you're having wifi router problems that you can't correct on your current router you have three options.
1) Try that WiFi dongle with the understanding that you might not be able to return it if it's not compatable.
2) Get a new, compatable router
3) Get the Wii LAN connector and run a Cat-5 cable from your router to the Wii (that's what I did since our place was prepared for LAN when built (empty conduits run to every room from a central hub where we put the cable router) and the Wii was on the ragged edge of our WiFi coverage). Very effective and the connection will never drop but you may have to put up with cables running under your rugs, along the baseboards, etc. depending on how far your Wii is from the router.
Good luck and I hope this helps.