What would happen is that SAS would try to reroute you. First through IAD/EWR/ORD on SK + UA metal to SFO. If that doesn't work out (no availability or schedules not permitting), the next alternatives would be LH via FRA/MUC + UA connections to SFO (unless booked on a direct LH flight to SFO). If these still don't work it's usually a night at an CPH airport hotel (do they use the fab Hilton or something less appealing?) and the same options the next day.
You coulde try to argue your case to get rebooked on alternative carriers (BA, AF, FI, DL, CO etc.) but that's a hit or miss - with no status and cheap tickets chances are quite slim. Any *A routings are always preferred.
Compensation depends on the reason for the delay that caused the misconnection. If it is weather-related, then SK doesn't really need to pay any compensation nor hotel nights and meals in the case of having to overnight. However in practice they usually do. As for compensation money for the delay it depends on the amount of total delay - if you would get rerouted and arrive in SFO with only a few hour delay then you don't really get any. If you get rebooked on the same flights the next day (i.e. a 24 hour delay) then some could be due - but you need to claim them directly from SK afterwards.
If this really happens to you the best thing to do is to be prepared: have a list of alternative connections to suggest to the customer service representatives, be active in seeking alternative solutions - don't simply take whatever is offered if you have a better idea and be proactive. It's better to call the call center as soon as you learn out about a potential misconnect to ask for protection on alternative flights than to miss your connection and wait in line for the customer service desk.