Hi Edith!
MS Internet Connection Sharing or ICS is mainly used for sharing a single internet connection with multiple computers on a network. What the service does is act like a gateway to the internet for other computers inside your network to connect through.
If you only have the one computer... you shouldn't need ICS turned on, for any reason... it shouldn't have to be turned on for you to access the internet. I'm kind of perplexed as to why you needed to allow that service acccess through ZoneAlarm before it let you on the net.
Which version of Windows are you using and what updates/Service Packs have been installed. If you are using WinXP with SP2 installed, is it possible you have both the Windows firewall and ZoneAlarm active at the same time? I could see this causing conflicts if it is the case. If it is... turn off the Windows firewall... ZoneAlarm is a more robust solution.
Now, the programs asking for access... hard to say whether they are good or bad with having a better description of them. The ones that are just a bunch of numbers don't sound right to me and could possibly be an infection. You may want to download
HijackThis and post a log to the
HijackThis Logs forum. If you do post a log, make sure you put HiJackThis in its own folder so that it can create the backups it needs to. Also make sure you post the complete log, unedited, header info and all.
As for powering down your cable modem... dont worry about it. Shouldn't have to turn it off. If you're going to power something down, just turn your computer off. If you don't feel like turning it off and you want to shut off access to/from the net while you are away from your system, you can lock it down with ZoneAlarm. Right click on the ZoneAlarm icon in your system tray(bottom right of the screen) and choose "Stop all internet activity". You can enable it by right clicking on the icon again and choosing "Resume internet activity" or something of the sort.