They will continue to ripen after being picked, to an extent. If they're picked at about 50% of the final color change they'll usually fully ripen, so I think it's a good idea to pick them at this stage to get maximum production and still get fully ripened pods. Unless I'm saving the seed, then I would let them fully ripen before picking...
Well, they do change colors... but I'm not sure that I would call an off-the-plant ripened pepper fully ripe. The second you cut the stem from the plant, the fruit is frozen in time in some aspects.
In my opinion picking pods under-ripe and then letting them ripen will not produce the same great flavors as vine-ripe so it would be up to the individual to decide what best to sacrifice for their personal wants/needs
Yep. I want the full sweetness and "complete" ripe flavor, which is why I tend to not ripen any peppers off the plants. I normally make sure that the peppers have at least reached their "final" ripe color, but I go a bit further and have them sit there for another week or so or until they've darkened a shade or so. Of course, this is the opposite of what you want to do if you are more interested in just getting more peppers, but it's a sacrifice I am usually willing to make.
Of course, this is not always the case though... sometimes I'll pick a pepper only a day or so after it has fully turned its final color, sometimes I'll try one on its second out of three colors (for those that have three color phases), and on occasion I might even pick one green for the hell of it. It really just depends on my curiosity, how anxious I am (to try a new variety), what I'm planning on doing with it, and various other things...
I pick 'em when I want 'em (which is usually when fully ripe), but one thing I'll only do under extreme circumstances is try to ripen them off the plant; I'm more likely to just freeze them as-is, really (which I have done).
The bottom line is, when you pick your peppers is really up to you. I think everyone's got different ideas, and probably stray from their usual timing from time to time...