The Numex Big Jim is a hybrid. Thus, there is no point at all in saving the seeds. Sorry
I would not let them ripen to red. They taste pretty good green, but are actually kind of bleh when fully ripe. Just my opinion, ymmv of course.
Big Jim is not a hybrid but as you mentioned, your plant could have been crossed or hybridized since it wasn't isolated, but chances would still probably be in favor of pureness
As for saving known crossed seeds, you better be prepared to grow out a lot of plants for a lot of years if you're trying to stabilize the variety, and be prepared after the f1 generation to have all very different plants for many years. Experimenting is always fun but its just so much easier starting with seeds known to be true or known f1's
Big Jim is not a hybrid but as you mentioned, your plant could have been crossed or hybridized since it wasn't isolated, but chances would still probably be in favor of pureness
As for saving known crossed seeds, you better be prepared to grow out a lot of plants for a lot of years if you're trying to stabilize the variety, and be prepared after the f1 generation to have all very different plants for many years. Experimenting is always fun but its just so much easier starting with seeds known to be true or known f1's