Did Jupiter really make Earth (in)habitable?
Perhaps, but it's tough to tell.
I wrote up an answer for a question on Physics getting at some more specific issues here; bear with me if I re-use a little bit of it.
How is it physically possible for such a migration to occur?
Planetary migration in the Solar System is described by the Nice Model (or, actually, the Nice II Model, its successor). Basically, the gas giants started out at different orbital semi-major axes than they are in today. There is one change in order, though: Neptune is closer in than Uranus (making the order Jupiter, Saturn, Neptune, Uranus). Also, there may have been a 5th giant (actually an ice giant), which was subsequently ejected. It would have been similar to Uranus and Neptune in composition and size.
Here's a quick summary of the Nice Model:
- Planetesimals and Kuiper Belt Objects interact with the giants, moving Jupiter inwards a tad and the other three (or four) outwards.
- Eventually, resonances come into play. These are ratios between orbital periods. Jupiter and Saturn cross a mutual threshold.
- Saturn moves outward and Jupiter moves inward. A chain of events happens, where Neptune and Uranus are dramatically moved outwards, with Neptune being chucked to the rear (and the hypothetical 5th giant being ejected). You can watch the video given in that question here.
Here are the before and after pictures,
stolen borrowed from that question:
I can, specifically, recommend the papers I mentioned there: Fassett & Minton (2013), Deienno & Nesvorny (2014), Brasser et al. (2009), Nesvorny (2011), Nesvorny & Morbidelli (2012), and Gomes, Levison, Tsiganis and Morbidelli (2005) - in no particular order, I think.
Is there a way to verify the migration hypothesis with observations?
Well, simulations were used in some of the mentioned papers, especially in those co-authored by Nesvorny, who was involved in recent developments of the Nice Model. These lead to the hypothetical 5th giant, because in some cases, it is necessary as a sort of sacrificial lamb, to prevent Uranus or Neptune from being flung out.
We really can't observe this happening in real time, so simulations are our best shot.
How different would have been the hypothetical planets in the area currently occupied by Earth?
The Late Heavy Bombardment might not have happened. That's a whole avenue of investigation into alternative astronomical history. It would involve fewer craters and possibly less fewer small collisions between some objects. It would be a calmer place. No migration = no LHB.
Beyond that, it's hard for anyone to say. Detailed simulations are really difficult, because the exact conditions that matter - not just orbital positions - are hard to model.
If such a migration toward the Sun started, what prevented Jupiter from continuing inward and collapsing with the Sun?
There just wasn't enough impetus from the encounter with Saturn to fling it inwards. A migration isn't just a planet rushing willy-nilly into space; it's a delicate system of slight orbital changes. All odds were against something like this happening.
Is it possible for such a migration to occur again?
Only if the right resonances appear, which is unlikely.