Does watching something on the network website count towards any of the rating? I know this doesn't apply to SG, but what about Hulu for the shows that are on there?
Absolutely it counts. Networks frequently boast about online viewership.
Some CW shows accrue more viewers through their website than their broadcast channel. Like On Demand, advertisers often have a captive audience (no fast forward or skipping commercials) for a TV series shown via a network's website, which nearly guarantees an ad will be seen.
With Hulu, I think viewership determines the price that a content provider can charge Hulu for streaming privileges. More viewers means a higher price tag, which means a more profitable show and a greater chance for renewal.