First some general preamble not specifically related to your question: the description of the site from the first page of the tour is:
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics and astronomy.
I interpret this fairly loosely because I consider everyone who wants to learn about physics a student regardless of whether they're in a formal course or not. So for me the key issue is whether the OP is interested to learn and is willing to put in the effort that learning requires. If I think they just want some gee whizz facts with which to impress their friends I'm likely to ignore the question.
Now let's consider the accusations you level. Your statement:
If the community cannot be 'bothered' to calculate anything or offer up answers, then why not reframe it into a 'we'll guide you to answer your own questions' site instead?
strikes me as petty and ill-informed. Every day I see busy working physicists putting in considerable amounts of their time for free to answer questions. It's all very well for me because I'm retired and have free time to burn, but I see answers from people who have to work (as physicists) to pay their mortgages or to pass their exams. To dismiss the contribution these people make with the phrase cannot be 'bothered' to calculate anything or offer up answers is exceedingly ungracious.
Now let's consider your question. I think it's a pretty good question, but the first part of it is one that I would expect anyone interested in physics to be able to answer themselves. After all, you just need Newton's law of gravity and the radii of the planetary orbits, which you can get from Google. If I were presented with this in a tutorial I would require the student to answer it themselves and I would provide hints only as required. Rereading the comments now, CuriousOne responded more or less exactly as I would in a tutorial. Admittedly that's in a formal teaching context, but then I tend to view the Physics SE as an informal teaching context and the same rules apply. So in this and similar circumstances I think it's entirely reasonable to make comments to the effect that you should answer the question yourself because that will benefit you more than just answering the question. Maybe that sounds a bit patronising, but isn't any teaching essentially patronage?
The second part of your question strikes me as fascinating and bloody difficult. In principle you can just take the orbital periods and current positions and try to calculate when the planets will line up. However one of the points in Schwern's answer is that the Solar System is dynamic and not just some clockwork mechanism, and there may well be effects that prevent a near perfect alignment and will gently perturb the orbits to avoid it. I didn't attempt an answer to this because I can't answer it!
And finally some footnotes that are entirely my personal opinion and unsupported by logical argument. What motivates me and others to put in so much effort isn't entirely clear to me. But then what motivates scientists in general isn't obvious - it's certainly not the money :-) But what demotivates people is clearer, and being taken for granted is one of the things. If I'm going to put in the work to answer a question I expect a reciprocal input from the student.