I’m pretty certain that’s not significant. The sun is by far — by far — the dominant source of external energy from the earth’s point of view, and the principal driver of our weather. In particular, the sun’s irradiance on earth is about 1300 W/m^2, whereas the cosmic background radiation is about 3e-6 W/m^2 — a ratio of about 400 million.
(Can’t address your second question I’m afraid.)
If that other stuff is going fast enough in other directions initially, especially if it’s far from the Sun, the Sun’s gravity will be extremely weak. The other material might fall towards it, or it might not. I think this is a bit like the dilemma of whether the Universe has enough matter to fall back in on itself or enough kinetic energy to keep expanding forever.
Read The Nine Billion Names of God by Arthur C. Clarke.
A short story along the lines of your OP.
Well, they’d still have each other. 
Objects travelling at solar escape velocity or higher — which, at distances of many light years, would be an easy threshhold to meet — would enter a parabolic or hyperbolic trajectory around the sun. Even objects with speeds below escape velocity would most likely enter huge elliptical orbits. So, we would never see the vast majority of these objects. They wouldn’t come anywhere close.
A lucky few would just happen to be headed in this direction, or be moving slowly enough in the sun’s reference frame that they’d begin falling directly toward us. Of course, by “few” I mean a tiny percentage, but then again, that’s a tiny percentage of a lot of objects. So I suppose we would have something to worry about, eventually.
It would take an awfully long time for any of them to reach us however. The sun itself will die long before then, and we’ll have more serious troubles to deal with.
No doubt about that. But it still doesn’t change the fact that the ambient temperature just got 3 degrees cooler assuming that the sun keeps on shining as brightly as it was before. I would assume that the temperature change would probably be too small to have any effect on the power output of the sun. Since the heatsink that surrounds us just got colder, the earth would eventually cool off by three degrees, too.
To amend my own reply to mangeorge…
Some “back-of-the-Starbucks-napkin” calculations give me about 70 million years for an object 4.2 ly away from the sun, initially at rest, to fall into the sun. At the distance of the Andromeda galaxy, the fall time stretches to a whopping 2.5*10^16 years (assuming away no other important effects, like the expansion of space). But the further you go out, the harder it is to find objects close enough to “at rest” that might be candidates.
Electronic orbitals and such might be disturbed by a sudden drop in the charge density of the universe. This could have nasty consequences for life on earth, or even the stability of the remaining atoms.
I’m afraid it doesn’t work like that. You can’t just add and subtract temperatures like this:
TempAfter = TempBefore - 3
For blackbodies, the energy involved goes as the fourth power of the temperature. So it’s more like this:
TempAfter[sup]4[/sup] = TempBefore[sup]4[/sup] - 3[sup]4[/sup]
If TempAfter is, say, 300K, then TempBefore is 299.9999993K, not 296K.
I meant 297K at the end, not 296K. Subtracting was never my strong suit.
Such an event would have to have an effect on the space-time continuum. There’s no way you could obliterate so much mass and energy without some effect.
A warp maybe? In the fabric of time? Our SS goes shooting through a wormhole to somewhere else, but there is no somewhere else?
Peace,
mangeorge
To this, I have to say (where’s the tongue-in-cheek smiley?), no.
They would obviously be hiring more astronomers to figure what the hell happened to Proxima Centauri (and 14 more stars in the next 7 years), and whether Sol is just gonna blink out unexpectedly in the same manner.
Hmmm, in retrospect, that doesn’t make a lot of sense, does it? If you could just add thermal sources like that, two 300 degree flames would heat something to 600 degrees. Looks like it would have very minimal effect, then.
I really need to learn to stop and think for a few hours before I make any statements about thermodynamics 
The disapperance of all the stars could be arranged by a malicious demon, who (for reasons of his own) could have stolen all the stars in the universe through a series of wormholes;
if he started with the most distant he could concievably arranged to make all the stars disappear as seen from Earth ** all at the same instant**…
thiswould have a profound impact on theology, for starters.
I am a little surprised that no-one has mentioned Mach’s principle;
http://www.geocities.com/jefferywinkler/machsprinciple.html
the distant galaxies are supposedly responsible for the observed phenomenon of inertia.
I have no idea if this principle would apply, but unless some obliging demon does perform this rather extreme experiment, we may never know.
SF worldbuilding at
http://www.orionsarm.com/main.html
Darn, I was going to show off by mentioning Mach’s principle but I see that the previous poster had this idea as well.
Anyway, it is possible that inertia would cease to exist with the absence of stars. That would have HUGE implications. On the other hand, even without stars we would still have the quantum sea of virtual particles so perhaps inertia would still be with us and we wouldn’t notice any changes due to the absence of all that matter.