How do people cross fast flowing rivers without a dedicated wading staff?
I can answer this only for Colorado and California. You find a place where you can cross safely, given your strength and balance. This may mean a significant detour.
First, for below timberline: Scout upstream and downstream for a log that has fallen across the stream. You will eventually find one that you are able to cross on. You may have to walk on the "wrong" side of the stream longer than you really want to, but be patient. By wrong side of the stream, I mean the side where the trail isn't; it's probably overgrown and/or hilly and hard going. But you will find a log.
Second, for above timberline: Walk upstream past several confluences of minor streams with the main stream and eventually the main stream will be crossable. If this is a longer detour than you really want to make, be patient!
On a day hike, never cross a barely crossable snow-melt stream in the morning unless you know of a much easier place to re-cross it in the afternoon. Snow-melt streams can enlarge dramatically during a day of melting.
As for how to cross that stream in Iceland without a wading staff if one is alone, the OP did not ask that question and I don't know how to answer it. If there are several people, see this question, especially the answer by Lagerbaer If I have to cross an icy, flowing river, what are some ways I can cross safely?
As for swimming the lake, as the OP contemplated, unless you know the temperature of the lake and the temperature you can tolerate in a swim, it seems risky. See Swimming to Antarctica by Lynne Cox. Lynne, a well insulated young woman specializing in long, cold swims, swam across Lake Myvatn in Iceland as part of her training to swim the Bering Strait from Little Diomede to Big Diomede. Lake Myvatn was 45 degrees F, and she found it cold when she started her training.