1. EU family reunification permit.
2. Probably yes as you would have to show you could support both of you if she's no longer working.
3. There are plans being made to limit immigration from the EU countries as per the recent vote by the Swiss public. If it actually happens then getting a permit even as an EU national will become difficult. One of the options being discussed is quota limits which would also apply to family members, which isn't the case at present. So it's better to do it now before the law changes.
4. Doesn't make any difference I don't think. If you get paid as a PhD student you get paid regardless of the permit type. Students are limited to 15 hours a week work-wise.
These may help: https://www.ch.ch/en/family-reunification-eu-efta/ https://www.bfm.admin.ch/content/dam...nnachzug-e.pdf
Be warned as an American citizen you're still required to file US tax returns no matter where you live in the world and could owe the US tax on top of Swiss ones. Any bank account you open you will need to sign a W-9 form for so the bank can pass the account info on to the IRS. You will also need to report any account/s with an aggregate total of over $10,000 at any time of the year on a FBAR form. So keep your accounts totally separate from your wife's otherwise hers will have to be reported too and she would also have to sign a W-9.
Start your research here: http://www.irs.gov/Individuals/Inter...-Aliens-Abroad