A recent question asked about applying for tenure-track jobs before the end of a postdoc. I'm in a similar position, also in math, and would like to ask the same question with an additional twist.
I'm in the middle of a postdoc. I've been advised to apply to places I'd definitely accept offers from this year, but my wife will also be looking for a job. She is not quite looking for an academic job, but the ideal outcome would be for her to work at the same institution as me, in a non-faculty job. (Let's say she wants work in the registrar's office.) The hiring for her sort of job doesn't take place so far in advance.
I would be very reluctant to accept a job a year early if I did not know for sure that my wife would also be able to find something there, especially if we were not moving to a big city (and we would prefer not to, if we can help it).
I'd be grateful for any general wisdom, but here are a couple concrete questions:
If I get an offer and turn it down because she can't find anything, and finish out my postdoc instead, would this be a very bad move?
If I get an offer and she can't find anything, is it possible I would be able to defer a year to help her job hunt?
I'm a little confused about your thinking on this matter: given that hiring for the sort of job your wife has generally happens on a shorter timeline, how could you ever know that your wife will have a job when you have to make a decision about an offer? I think you may just have to go for it and hope for the best. You can always apply for jobs again if things don't work out (I've done this twice); if you can afford to be choosy about jobs now, then likely you can get another. The one wildcard is that if you get an offer, you may be able to negotiate a job for your wife before you accept. A non-academic position is an unusual request, but on some level its a smaller ask than, say, a TT position for her. The situation might be clearer if you said more about your wife's experience and skills. About your specific questions:
Well, it's not a good move. Probably the interview and offer process will move too quickly for your wife to reasonably expect to have found a job for six months hence, and people in the department might reasonably ask why you applied and went through the interview stages if you weren't willing to take the risk of moving there.
If you make the decision before you accept the job, this is a reasonable (and often made) request, but probably not after (and I'm a bit confused as to why you would ask). As with 1., the timeline on TT job offers (often 1 week) is too short to really know anything about her job prospects, so I assume you mean after accepting. Negotiation about start dates should happen before you accept the offer; of course, you can try to organize it after, but that is more likely to be received poorly (of course, it all depends on the situation, how much they are counting on your teaching, etc.), and you've lost all your leverage. Given that your wife's job search will be easier once she's in the location (and can network, etc.), it will seem like an odd request and will likely give your colleagues the sense that you already have one foot out the door.