The short answer: it depends on the length of your (the J-1's) stay. If that is not longer than 30 days, you should not have any problem with the J-2 remaining in the country (see @Dorothy's answer).
If your stay is (intended to be) longer than 30 days, the J-2 cannot stay (see @Dorothy's answer), but one question is whether the J-2 can at least stay for the first 30 days of your longer-than-30-days absence. Considering that you usually have to complete a notice of absence, an out of country request or anything similar for stays longer than 30 days, I understand from the wording of the out of country request at University of Washington that the J-2 has to leave at the same time as the J-1:
We certify that:
- J-2 dependents will depart the U.S. with the J-1 Exchange Visitor
Regarding your specific options:
I go to France alone for a couple of weeks, go back, and then we go to France together for three months.
This should be fine, as long as a couple of weeks is not longer than 30 days.
I go to France alone; my wife joins me there a month later; we return together three months later.
This will be a problem if your three-months stay is intended to be that long from its beginning. The J-2 has to leave with you, if my reasoning above is correct.
I go to France alone; my wife joins me there a month later. While there, I learn that I get a job. I resign from my U.S. position, cancel my visa and neither of us returns to the U.S. in the fall.
From an immigration point of view, this is no different from the previous option. The J-2 has to leave with you, and no one will care whether you ever come back or not. I see no problem with resigning from your position; however, I generally wonder whether a non-completed J-1 program might jeopardize future J-1 applications. You might need to figure that out with your sponsor, but it should not make any different whether you do this while in the US or not.
The one open question is what happens if you leave for an intended period of time not exceeding 30 days, and don't return as planned. It would not be credible for me to speculate what exactly happens, but I guess there can be no doubt that you need to inform your sponsor (compare the notice of absence: "I will inform ISSS and my department if my schedule changes."), and that the J-2 needs to leave as soon as your sponsor determines that your stays exceeds 30 days. I do not know whether that counts as overstaying on her part; nor whether that impacts your ability to return into the program if you fail to find a job in France.
Note that you are planning to "spend a few months in [your] home country for job interviews and vacation (while keeping [your] legal J-1 status)". This would involve convincing your sponsor that your absence is program-related (compare, again, this notice of absence). If you fail to do that, there is a possibility that they terminate your J-1 program, which means that you and the J-2 have to leave, maybe immediately, but certainly without a 30-day grace period (which you only get when you complete the program).
This is not legal advice, obviously.
Adding to @Dorothy's answer, I have found this on the webpage of University of Minnesota:
Please Note: It is unclear how the status of a dependent visa holder is affected if the dependent remains in the U.S. when the principal visa holder is temporarily absent from the U.S. It is recommended that the J-2 not remain in the U.S., if the J-1 will be absent for more than 30 days.
So even people having educated opinions (such as those at that particular International Office) may not know the exact answer.