Let’s say Q decided to rid our solar system of all the other planets (they just disappear instantly), how would their absence affect Earth, if at all?
Really, all it’d do is make the night skies a little less interesting. The only things in the Solar System which have any measureable affect on the Earth are the Sun and the Moon. Well, and any object which happens to collide with the Earth: They have an effect, too. But none of the planets are likely to do that any time soon.
Well, mythology would suffer greatly, and it’d probably be disastrous to mechanics (and thus to all modern mathematics and physics) not to have had such nicely-behaved toy systems to observe.
IIRC Jupiter has a small but measurable influence on Earth’s orbit – if any of the other planets do, it’s so miniscule as to be not worth mentioning, but Jupiter’s enormous mass and relative propinquity do cause a tiny but significant change from what Sun/Moon alone would do.
I seem to remember hearing somewhere that other planets, Jupiter in particular, are extremely useful in absorbing meteors and such. IIRC, Jupiter was called the solar system’s “Vacuum Cleaner”. The point being that we’d be kinda screwed without it.
Any astronomy buffs care to opine on that?
Astrologers would go out of business overnight (unless they predicted everything in advance, of course) 
Google is my friend.
Googling “Jupiter vacuum cleaner” turns up quite a few references.
http://www.solarsystem.org.uk/jupiter/
http://www.space.com/reference/jupiter/overview.html
Sounds like it was a good thing to have Jupiter around all these years.
“Don’t it always seem to go/
You don’t know what you’ve got till it’s gone” 
Nah, they’re probably make their predictions based on the movements of turtles or sand grains.
Most people would still fall for it. 
Probably be a lot more accurate …
Yeah, yeah, but what has it done for use lately?
Well, it did absorb the comet Shoemaker-Levy 9 in 1994. But I don’t suppose that would have collided with Earth anyway.
Most orbits tend to be rather ergodic in the long run. Eventually it would have hit us. Jupiter’s just so big that it’s more likely something’s going to hit it first than us.
Well, in certain circumstances, fortune-telling based on the movements of turtles would be scientifically accurate. Discworld, for example!
We would not have suffered the depradation of all the various Martian assaults on the Earth from the War of the Worlds througfh Mars Attacks (to say nothing of our losses because Mars needs women).
Well, I guess the annual Solar System Ball would be a real bummer. No dance partner for Mother Earth. I mean, geez, shew would have to take the Moon, again!
Then all the psychology book aisles would be devoid of good books such as Women Are From umm, umm / Men Are From umm, umm
Aha! A benefit!
“Razed orbit paths/
Here comes a great big rock.
Oooooh bup bup bup bup”
My Very Excellent Mother Just Never Served Us Nine Pizzas, because there would have been no reason for that mnemonic to be invented.
This was rather the point of my post. We did need Jupiter at the start of the Solar System, to clean up all the debris (unless that was all just magicked away, too). But now that most of the debris is gone, if it were to disappear now, no big deal. There are still some comets and other risky objects, and most of those will eventually hit something, but even then, if the Solar System were devoid of all objects save for the Sun, Earth, Moon, and comets, I suspect that more of those would hit the Sun than the Earth. In fact, even with all the planets in place, falling into the Sun is still by far the most common cause of death for comets: As I recall, about one comet a day ends up hitting the Sun.