If you can ping any outside IP then you have access to the outside world. Not all IP's are pingable even though you can connect to them.
I have 6 machines on my network (at the moment, though about to retire a couple of them) for various things. I have WWW, IRC, FTP, MS Exchange all experimental stuff and all reachable from the 'Net. If you ping my IP you won't get a reply though. Maybe there's no real advantage to it but I do like it that way.
You say you don't use the router for ADSL... Do you still connect with just the one machine or do you try to connect several using one machine as a gateway/hub? If you do network probably best to use the router as these are specifically designed for this. ADSL is weird and can take time to play around with if you use a PC as a gateway. You'll get it going but sometimes you have to alter what is known as MTU settings. This can be done in the System Registry though only go here if you know exactly what you are doing. If you stuff up registry settings it may render Windows useless. See this site for more details
http://ozcableguy.com/The standard MTU settings can 'choke' the bandwidth in the LAN PC's. mIRC will sometimes connect whereas trying to browse websites can be all but impossible. Cable and dialup arn't affected as ADSL is.
If you are just using one machine I'd be making sure the modem is okay. I recently had to replace mine as it'd been on for the last two years and done a splendid job. Nothing lasts forever though and broadband modems are not always resilient objects.
