Hi Voodoo,
i) I broke my leg, a situation that forced me to stay home. (appositive)
(ii) I broke my leg, forcing me to stay at home. (,+ing form - modifying the act of breaking the leg)
(iii) I broke my leg, and I stayed at home (I am using ,and form here just to prevent it from being "run-on sentence" ) I am a little bit confused.
Actually, all three of the sentences you suggest are fine. The third one doesn't communicate the exact same thing, because it doesn't express that the fact that you broke your leg is WHY you stayed at home; breaking your leg and staying at home could just be two totally unrelated things you happened to do that day. But grammatically, they're all fine (and logically, they're all fine, too -- (i) and (ii) just have a different connotation from (iii)).
Let's look at the original sentence again for a minute (I've chopped off the very beginning, because it's irrelevant here):
The way for a merchant to generate the most profit was to move a limited amount of scarce goods to San Francisco as quickly as possible, rather than
to carry larger loads more slowly, determining the design of the clipper ship.
I wouldn't say that the problem with that option is that it's a run-on; I wouldn't even call it a run-on. It's just that here, it's ambiguous just what "determining" is modifying, since there are so many verbs that precede it -- it could be modifying the
fact that the way to generate the most profit
was to..., or it could be modifying the verb
"to move" (i.e. if the meaning were something like "The best strategy was to move a limited amount of goods, determing the design of the clipper ship as you did so" -- logically wacky, but grammatically possible)..., or it could be modifying the verb "to carry" (i.e. if the meaning were something like "The best strategy was to move a limited amount of goods, rather than to carry large loads while determining the design of the clipper ship" -- again, logically wacky, but grammatically possible). Inserting "a situation" and transforming the modifier into an appositive gets rid of all that ambiguity.
In your sentence (ii), there IS no ambiguity, because there's only one thing that "forcing" could be referring to, so your sentence doesn't suffer from any of the confusion this other sentence does. So I don't think you'll get trapped
