Yes, within 3 months of arrival, and it runs from the date of arrival

You can get your permit without it, but the local government office (Gemeinde or Kreisbüro) needs a copy of your policy when you get it.

It doesn't matter for your apartment hunting.

To get prices, check our www.comparis.ch. This is commercial, but still has good info.

You may get a contribution towards the cost from your employer (Credit Suisse 30 or UBS 50 etc).

FYI, we both have excesses of CHF 2500 to keep costs down

And just search this site for "health insurance" to see other posts.

Edit:
An insurance company cannot reject your application for basic insurance - they must accept you.
They can insist on a questionaire for any supplementary insurances.

You don't need accident cover if you work more than 8 hours per week - the employer takes care of that.