I think there's a substantial chance that your Internet slowness and your sites being hacked were two unrelated problems that happened to occur at the same time.
Since resetting your router took care of the local Internet slowness, it implies that that was being caused by the router, not by your PC. If that's the case, do you by any chance have a cheap consumer router (i.e. Netgear, Linksys, D-Link) that's more than a couple years old? If so, were you using BitTorrent? There's a known issue with a huge number of consumer routers that cause them to bog down to hideously slow if you run BitTorrent through them for more than a few hours. Stopping BT does not fix the problem -- only resetting the router or waiting an absurdly long time (24-72 hours) will fix it. If this was your problem, the only way to prevent it from happening again are 1.) don't use BitTorrent, 2.) get a new router, or 3.) upgrade your router's firmware with a 3rd-party firmware like DDWRT or Tomato.
As for the sites getting hacked, WordPress does have vulns discovered in it pretty often, so it's important to keep WP sites very up-to-date if you have them. However, it's quite possible that someone rooted the box at 1and1 through some other guy's site on the same server, then compromised every site there (this has happened to my installations before, when I used shared hosting.) The only way to prevent that is... to not use shared hosting. VPS or dedicated doesn't generally have that problem. A less lame hosting provider might help, but even if your hosting provider is great, as long as it's shared you can bet somebody is running a crappy, insecure site on it somewhere, and no hosting provider keeps everything patched and up-to-date all the time (to do so would require frequent reboots, and hosting companies know nobody likes their sites being down.)
If your PC really was infected by something (I'd scan for worms/viruses/zombies with Kapersky, then spware/adware with Spybot S&D), then the only way to be truly sure it's clean is to reinstall the OS ("nuke the facility from orbit, it's the only way to be sure.") That may be overkill for your average user, but if you're running a business off your PC... you want to be sure. The last thing you want is to change all the passwords on all your sites, only to find your PC is secretly emailing them to some dick in Russia. On the other hand, if Kapersky & Spybot say it's clean, the problem was probably the router, not the PC.
And finally, if your hosting provider allows you to disable FTP and use only SFTP/SSH... it's a good option to turn on. FTP sends your password in clear text over the Internet every time you log in.