We had no option about Archie's leg. It was too smashed to be saved. So we've never had to make the determined choice.
That said, you can tell by reading all the stories from everyone on here, that they get on just fine on three legs. You cat may be a bit more unsteady due to lack of tail.
As for the crying, Archie did. Probably a mix of fear, confusion and pain. It must be hard to lose a leg! But he soon got over it.
You mustn't feel guilty. Accidents happen and I do believe that cats should be allowed outside