Re: what how would earth be different if the sun had a wide double-star system?

It depends on how far about the stars were. I tried looking up 'wide binary' in a half-dozen astronomy texts, and none of them defines the term formally. And when I asked the other graduate students, everyone knew the term, but no one would commit to a definition, either. So I'm going to be a bit more general in my answer, and if you know how far about wide binary stars must be, you're welcomed to ignore the irrelevent bits.

You did say 'wide' binary, so I will assume that the companion star is at least 10 Astronomical Units (or AU - the distance from the Earth to the Sun, as you probably know) away from our Sun. This is thousands of times the radius of a typical star, so I think of these guys as pretty far apart. Anyway, if it were around this close, there would be no outer solar system at all, since the companion star would be cruising right through it. In fact, there would be no inner solar system, either, since the orbits are all unstable. So this would be a boring system to look at.

Well, what if the companion where 100 AU from the Sun? Well, the outer planets would certainly be in trouble, still. The star would be close enough to them, and they are far enough from the Sun, that their orbits would still be played with pretty heavily. In fact, they might not have been able to form at all. Earth, however, would be somewhat safer. I can't tell you that we'd be OK here, since there is a lot of uncertain in planet formation models, but I wouldn't be surprised if we were pretty much OK.

Moving the companion star farther out again, to 10,000 AU. This is the Oort cloud, so the companion star would probably clean our the Oort cloud pretty soon after the solar system formed. In a way, this would protect us on Earth, since there would be no Oort cloud comets to come crashing down on us and wiping out much of life on Earth (as with the dinosaurs 65 million years ago). On the other hand, these periodic collisions and mass extentions make way for new life forms, so we humans would probably not have ever gotten the chance to evolve. So it's a mixed blessing.

The planets themselves should be OK, since the companion star is so far away that the Sun's gravity is much more important than the companion star's contribution.

If the companion star were still farther away, say 100,000 AU, the effects become almost irrelevent. To convince yourself of that, remember that Alpha Centauri, the nearest star to our Sun, is around 200,000 AU away.

Up until now, I've been assuming that the orbits of the stars about each other are pretty near circular. But for a moment, think about what would happen if the two stars have highly eccentric (or elleptical) orbits. If the companion star were still 100,000 AU away on the average, but sometimes came as close as 10,000 AU to our Sun, we would see an occasional rain of comets down onto us, since the companion star would be passing through the Oort cloud once every orbit. The orbital period would be about 30 million years (from Kepler's 3rd law), so this wouldn't happen that often. But it would affect life's progress on Earth, since every 30 million years we'd probably find ourself struck by a large comet.

I'd love to point you to more literature on this topic, but I know know of anyone who has done much research in this area. We're still trying to fully understand out solar system, so we haven't really spent a lot of time addressing such interesting "what if" questions yet. Hopefully, in the next ten years we will be able to observe planets around other stars, including binary stars. Then we can start to really answer this question!